Page 1 of WEB CHAPTER 3- INTERSECTIONS
WEB CHAPTER 3- INTERSECTIONS
When two solid objects intersect each other, the line common to both objects is called the curve of Intersection. Intersection of two planes is simply a line but when a plane or two solids intersect, it is a curve. A numbering technique is used in getting a curve of intersection. First, draw orthographic views of Object 1 and Object 2 at their relative positions. Then follow the three steps given below: Step 1 – Projections from one view A. Divide the area of intersecting Object 1 in 12 equal parts and number as 1, 2, ….12. B. Take projections from these points along the axis of Object 1 up to the boundary of Object 2 and number these points 1’, 2’….12’ respectively. C. Take projections from the points 1’, 2’….12’ to the other view. Step 2 – Projections of other view A. Divide the area of Object 2 also in 12 equal parts and number as 1, 2, ….12. B. Take projections from these points along the axis of Object 2 up to the projection lines drawn in the step 1C. C. Find the intersection points of the same numbers from both the views and number them as 1”, 2”…. 12”. Step 3 – Draw the curve of intersection passing through the points 1”, 2”…. 12”. CAD In AutoCAD, the curve of intersection is drawn automatically when 3‐D solid objects are joined or subtracted. The curved surface of a solid model is shown by many contour lines called isolines. The following commands can be used to create 3D primitives at a specified location directly: ISOLINES Sets number of contour lines per surface BOX Draws a box of specified length, width and height SPHERE Draws a sphere of specified radius CYLINDER Draws a cylinder of specified radius and height CONE Draws a cone with circular or elliptical base of specified base and apex point Draws a wedge of specified length, width and height tapering along WEDGE the length axis Page 2 of WEB CHAPTER 3- INTERSECTIONS
TORUS Draws a ring of specified ring radius with circular cross‐section of specified radius. Solids can also be created by the following commands: EXTRUDE A solid is created by moving a 2‐D profile along a specified path. REVOLVE A solid is created by revolving a closed profile about an axis. SLICE Two solids are created by cutting a solid in two parts using a slicing plane. INTERFERE Creates a new solid from the common volume of two solids keeping original objects. These solids can be joined or subtracted using following 3D‐Editing commands by following Boolean operations: UNION Joins two or more solid objects SUBTRACT Subtracts one solid object from another INTERSECT Creates a new solid from the common volume of two solids like INTERFERE command but does not keep the original objects. W3.1 INTRODUCTION When two solid objects intersect each other, the shape of the line of intersection depends upon the shape of the intersecting surfaces, and their meeting boundaries get modified. When two plane surfaces intersect, their intersection is a straight line. When a plane intersects a curved surface or where two curved surfaces meet, the line of intersection can be either a straight line or a curve depending upon the type of surfaces and their relative positions. To draw this curve of intersection, at least two views are required in orthographic projections. Figure W3.1 shows a cone intersected by a cylinder. The curves AB and CD are the common boundaries of the cone and cylinder at the location of intersection. This chapter deals with the drawing of this modified boundary known as the curve of intersection. Page 3 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.1 Intersection curve of a cone and cylinder W3.2 INTERSECTION OF PLANE INCLINED SURFCAES Figure W3.2A shows the intersection of two plain surfaces inclined to each other at an angle. One of the surfaces ABCD is horizontal, which is being intersected by a triangular inclined plane EFG. Line 1‐2 indicates the line of intersection. To show these planes in orthographic view, follow the steps explained below: 1. Draw rectangle ABCD as the top view of the first plane. It is being represented as a horizontal line in the front view. Its side view is also a straight line. 2. Draw the other plane EFG of the given size at the specified angle and position w.r.t. the first plane in the side view. 3. Take its projections in the front view and mark them as EFG. 4. Points 1 and 2 are the locations where edges EF and EG cut the horizontal plane in the front view. These points 1 and 2 coincide in the front view. 5. Take the projection of points 1 and 2 in the top view as shown in Figure W3.2B. Join 1 and 2, which is the desired line of intersection. Page 4 of WEB CHAPTER 3- INTERSECTIONS
(A) Pictorial view of planes (B) Orthographic view of planes Figure W3.2 Intersection of two inclined surfaces W3.3 INTERSECTION OF SOLID SURFACES Although an object can have any shape, only some standard shapes have been selected to explain the method of drawing intersection curves. The method can be extended to draw any other shape. Even for standard shapes there can be many combinations but only a few are discussed in this chapter. They are: • Cylinder with plane (Section W3.3.1) • Cylinder with cylinder (Section W3.3.2 ‐ Examples 1 and 2) • Cylinder with cone (Section W3.3.3 ‐ Example 3 ) • Cylinder with prism (Section W3.3.4 ‐ Example 4) • Prism with prism (Section W3.4.1‐ Example 5) • Prism with cylinder (Section W3.4.2 ‐ Example 6) • Prism with cone (Section W3.4.3 ‐ Example 7) • Cone with prism (Section W3.5.1 ‐ Example 8) • Cone with curved plane (Section W3.5.2 ‐ Example 9) • Cone with cylinder (Section W3.5.3 ‐ Example 10) Page 5 of WEB CHAPTER 3- INTERSECTIONS
W3.3.1 INTERSECTION OF A CYLINDER WITH A PLANE A vertical cylinder is shown in Figure W3.3. This is being cut by an inclined plane X‐
Y. It cuts the top surface of the cylinder along the side AB. The curve of intersection of the cylinder and the plane is shown in the side view. To draw this curve, follow the steps given below: Figure W3.3 Intersection of a cylinder with a straight plane 1. Draw the front view of the cylinder of the given size. Draw the inclined plane X‐Y at the given angle and location as shown in the front view. 2. Draw the top view of the cylinder and divide it into 12 equal parts by drawing radial lines at 30° interval. Number the intersections with the circumference as 1, 2…12. 3. Draw vertical projections from each point, i.e., 3 to 11 on line XY and mark them as 3’, 4’…11’. Points 1, 2 and 12 do not cut for this position of the plane. Page 6 of WEB CHAPTER 3- INTERSECTIONS
Numbers on the back side of the cylinder coincide with numbers of the front side, e.g., 3 and 11. 4. Draw the side view of the cylinder. Transfer the projections of the points 1 to 12 from the top view to the side view either by Miter line or by again drawing a semi‐circle and dividing it in the same number of parts. Be careful about the numbering. Draw vertical projections of points 1 to 12 on the side view. 5. Take horizontal projections from points 3’ to 11’ on line XY towards the side view to cut the vertical projections of the side view at 3’’, 4’’…11’’. 6. Draw a curve passing through all the points 3’’ through 11’’ to indicate the curve of intersection. The distance AB in the top view and the side view is the same. The cutting plane can be straight or inclined. The procedure for drawing the curve of intersection is the same as that for a straight line. See Figure W3.4 in which a curved plane cuts a cylinder. Points have been marked only on the right‐hand side of the cylinder, as there is no intersection on the left‐hand side of the cylinder. Figure W3.4 Intersection of a cylinder with a curved surface Page 7 of WEB CHAPTER 3- INTERSECTIONS
W3.3.2 INTERSECTION OF A CYLINDER WITH A CYLINDER A cylinder can intersect another cylinder at the ends, in between, centrally or eccentrically. Figure W3.5 shows two cylinders of equal diameters placed axially at 120° to each other. To draw the curve of intersection, first draw the orthographic views of both the cylinders of the given sizes at their respective positions. Divide the top of the vertical cylinder at 30° interval and draw vertical projections from the top in the front view. Next, in the front view, divide the inclined cylinder end into the same number of parts and take projections parallel to its axis to cut the projections of the vertical cylinder at 1, 2, 3, etc. Join the intersection of respective points to form the line of intersection. Figure W3.5 Intersection of cylinder at ends Page 8 of WEB CHAPTER 3- INTERSECTIONS
If a cylinder intersects the other cylinder in between as shown in Figure W3.6, the procedure remains the same as mentioned above. If the cylinders are not of equal diameters, the curve of intersection gets modified, but the procedure remains the same. The arrows indicate the direction in which the projections have been taken. Figure W3.6 Intersection of cylinders at any place Sometimes the intersecting cylinders are not co‐axial, but there exists some offset. Such an intersection is shown in Example 1 in Figure W3.S1. EXAMPLE 1 (Intersection of cylinders of unequal diameters eccentrically) A vertical cylinder having a diameter of 80 mm and a height of 120 mm is cut by a horizontal cylinder having a diameter of 50 mm and a length of 120 mm such that their exes are offset by 10 mm in the horizontal plane(Figure W3.S1). SOLUTION Draw orthographic views of the cylinders. Divide the end of each cylinder in 12 numbers and mark them 1 to 12 as shown in Figure W3.S1. Take the projections from the top view to the front view. Find the intersection of respective points. Join these intersections by a smooth curve, which is the curve of intersection. Note that the curve of the back side is shown by dashed lines as it is not visible. It can be Page 9 of WEB CHAPTER 3- INTERSECTIONS
noticed that the curve of intersection becomes different for the front and rear side shown by dashed lines. Repeat the process for the right‐hand side also to get the other curve of intersection. Figure W3.S1 Intersection of cylinders of unequal diameters EXAMPLE 2 (Intersection of inclined eccentric cylinders) For the cylinders shown in Example 1, if the axis of the smaller cylinder is inclined at 60° with the vertical axis, show their curves of intersection. Assume that the bottom end of the small cylinder just touches the bottom of big cylinder. SOLUTION Front View Draw orthographic views of the cylinders. Divide the end of each cylinder in 12 numbers and mark them 1 to 12 as shown in Figure W3.S2. Take vertical projections Page 10 of WEB CHAPTER 3- INTERSECTIONS
from the top view to the front view. Projections from the small cylinder in the front view are inclined parallel to its axis. Find the intersection of respective points. Join these intersections by a smooth curve, to give curve of intersection. Side View Since the small cylinder is inclined, a curve of intersection appears in the side view also. Take horizontal projections from the front view to the side view to locate points 1 to 12 for the top face and also from the intersection curve. Then take vertical projections from the top face points to projections from the intersection curve. Pass a curve passing from the respective intersection points. Figure W3.S2 Intersection of cylinders eccentrically W3.3.3 INTERSECTION OF A CYLINDER WITH A CONE EXAMPLE 3 (Intersection of a cylinder with a cone) A cylinder having a diameter of 65 mm and a height of 85 mm is intersected by a horizontal cone having a base diameter of 64 mm and a height of 100 mm such that the center of the cone axis coincides with the center of the cylinder axis (Figure W3.S3). Draw the curves of intersection. Page 11 of WEB CHAPTER 3- INTERSECTIONS
SOLUTION Draw orthographic views. Divide the base of the cone in the side view in 12 equal parts and number them 1 to 12. Take the projections of these points on the base of the cone in the front view. Join these points with the lines converging to the apex (Figure W3.S3). . Similarly, mark the points of the base of cone in the top view also and join them with apex. It may be noted that the numbers which start from 1 on the horizontal line in the side view starts from 1 on the vertical line in the top view. Wherever the converging lines of the top view intersect the cylinder, mark them as 1’, 2’,…7’. Take vertical projections from points 1’, 2’ ,etc., to cut the converging lines in the front view. Intersection of lines with same number are marked as 1”, 2”.. etc. Join all the points such as 1”, 2”, 3”… etc., to get the required curve of intersection. Figure W3.S3 Intersection of a cylinder with a cone Page 12 of WEB CHAPTER 3- INTERSECTIONS
W3.3.4 INTERSECTION OF CYLINDER WITH PRISM It is easier to draw curves of intersection for prisms than for cones, because the width of prism remains the same for the entire length. Figure W3.S4 shows a vertical cylinder intersected by a horizontal square prism. It will be an easy problem if the prism cuts the cylinder at its central axis of the cylinder. To show the effect of eccentricity, the prism is shown cutting the cylinder eccentrically. EXAMPLE 4 (Intersection of cylinder with prism) A vertical cylinder having a diameter of 65 mm and a height of 90 mm is intersected by a square horizontal prism rotated 45° about its axis. The axis of the prism passes at 45 mm above the base of the cylinder and is offset by a distance of 7 mm with the axis of the cylinder in the horizontal plane. Draw its intersection curves. SOLUTION Draw the orthographic views. Mark the edges of the prism in the side view as 1, 2, 3 and 4. Where the prism edges cut the cylinder in the top view, number them as 1’, 2’, 3’ and 4’. Take vertical projections from these points to the front view and horizontal projections from the side view. Mark the points of intersection of these projections as 1”, 2”, 3” and 4”. Join these points to give the curve of intersection. If the prism is of a large size, an increased number of points can be taken. Page 13 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.S4 Intersection of cylinder with prism W3.4 INTERSECTION OF PRISMS W3.4.1 INTERSECTION OF A PRISM WITH A PRISM Procedure for drawing the curve is the same as mentioned in Section W3.3.4. EXAMPLE 5 (Intersection of a prism with a prism) A vertical square prism has a base of 50 mm and a height of 70 mm. Its base is inclined at 45° with the principle plane. It is intersected by another horizontal square prism with a base of 25 mm and a length of 100 mm. The diagonal corners of this prism are aligned vertical and its axis is offset in the horizontal plane by 6 mm from the vertical axis of the prism. Draw the curves of intersection. Page 14 of WEB CHAPTER 3- INTERSECTIONS
SOLUTION The method is the same as discussed in the sections above and the solution is shown in Figure W3.S5. It may be noted that since the surfaces are flat, the intersections are straight lines. The intersection of the back side is shown by dashed lines. Figure W3.S5 Intersection of a prism with prism W3.4.2 INTERSECTION OF A PRISM WITH A CYLINDER EXAMPLE 6 (Intersection of prism with cylinder) A five‐sided vertical prism having each edge of 55 mm and a height of 70 mm is being intersected by a horizontal cylinder having a diameter of 65 mm and a length 100 mm coaxially in the center such that the axis of cylinder is parallel to the edge of the prism (Figure W3.S6). Draw the curves of intersection. SOLUTION Draw the orthographic views. Divide the circular base of the cylinder in the side view and top view in 12 equal parts and number them as 1, 2…..12. Note the numbering scheme carefully. Take horizontal projections from points 1, 2 etc., in the Page 15 of WEB CHAPTER 3- INTERSECTIONS
top view on the faces of the pentagon. The point of intersection with the face be marked as 1’, 2’,… 12’. Take vertical projections from these points to meet the horizontal projections from the side view at 1”, 2”…12”. Join these points in order by full lines and dashed lines as required. Figure W3.S6 Intersection of a prism with a cylinder Page 16 of WEB CHAPTER 3- INTERSECTIONS
W3.4.3 INTERSECTION OF A PRISM WITH A CONE EXAMPLE 7 (Prism intersected by a cone) A square prism having each edge of 57 mm and a length of 125 mm is inclined at 45° about its vertical axis. It is cut by a cone at the central axis diagonally at the middle of the prism (FigureW3.S7). The base of the cone is 100 mm and the apex length from the base is 135 mm. The apex projects out of the prism by 70 mm from the axis of the prism. Draw the curves of intersection for both the ends. SOLUTION The method is similar to the previous examples. Its solution is shown in FigureW3.S7. Figure W3.S7 A prism intersected by a cone Page 17 of WEB CHAPTER 3- INTERSECTIONS
W3.5 INTERSECTION OF CONES W3.5.1 INTERSECTION OF CONE WITH PRISM EXAMPLE 8 (Intersection of a cone with a prism) A right circular cone has a base of 90 mm and a vertical height of 95 mm. It is intersected by a 180‐mm long square prism having each edge of 15 mm at an angle of 30 degrees. The diagonal edges of the prism are vertical The axis of the prism cuts the cone at the centre at 40 mm from its apex. Draw the curves of intersection. SOLUTION Its solution is given in Figure W3.S8. Figure W3.S8 A cone intersected by a prism Page 18 of WEB CHAPTER 3- INTERSECTIONS
W3.5.2 INTERSECTION OF A CONE WITH A PLANE EXAMPLE 9 (A cone cut by a curved plane) A conical object is made of two solids; a cylinder and a cone. The cylinder has a 90‐ mm diameter and a height of 16 mm. The cone has a 90‐mm diameter and a height of 95 mm placed over the cylinder centrally. Both the objects are cut first by a vertical plane that is 8 mm from the axis of the cone for a length of 38 mm and then by a curved plane with a radius of 50 mm as shown in Figure W3.S9. Draw its side view showing the curve of intersection. SOLUTION a. Draw the front view, side view and top view. b. Select some key points on the front view, e.g., corners, edges and some intermediate points like A, B, …. H. c. Draw horizontal projection lines from these points up to the side view. d. Draw vertical projection lines from these points up to the top view to cut the top view at A’, B’ …. H’. Points B’ and C’ lie on a circle of radius R1 , and D’ and E’ are on a circle of radius R2 in the top view. Take horizontal projections from points A’, B’ ….H’ up to the miter line and then from there to the side view through the vertical lines. e. Find the intersection of respective points A”, B”,….. H” from the projection lines from the front view and the miter line. f. Draw a curve passing through these points. Page 19 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.S9 A cone cut by a curved plane W3.5.3 INTERSECTION OF A CONE WITH A CYLINDER EXAMPLE 10 –( Intersection of a cone and an eccentric cylinder) A horizontal cylinder having a diameter of 33 mm and a length of 120 mm cuts a right circular vertical cone having a base diameter of 92 mm and a height of 90 mm eccentrically by 16 mm in the horizontal plane and 40 mm above the base of the cone as shown in Figure W3.S10. Draw the curve of intersection. SOLUTION a. Draw the orthographic views. b. Divide the base of the cylinder in 12 parts in the front and top views. Draw horizontal projection lines from these points in the front view. c. Draw horizontal projection lines from points 1’, 2’ …. 12’ in the top view. Wherever these lines cut the base of the cone, draw vertical projection lines from those points up to the base of the cone in the front view. Join these points with the apex of the cone. Page 20 of WEB CHAPTER 3- INTERSECTIONS
d. Find the intersection of the lines drawn in steps (b) and (c) in the front view. e. Join the intersection points by a smooth curve as shown in Figure W3.S10. Figure W3.S10 A cone eccentrically cut by a cylinder Page 21 of WEB CHAPTER 3- INTERSECTIONS
W3.6 SOLID MODELING COMMANDS A surface model covers only the outer shape with a thin shell having a surface area but does not have any volume property. A solid model is a solid between the outer surfaces. Curved surfaces are shown by isolines. These lines are the contour lines per surface on an object. The default value is 4 and the maximum value is 2047. To set its value at the command prompt, type the command and specify a new value. Command: Isolines ↵ Enter new value for isolines <4>: 32 ↵ Type the command and press Enter. Specify a new value (say 32) and press Enter. On the Menu bar, click View and then from the pull‐down menu select Toolbars…. From the Toolbars dialog box click on the Solids check box. Close the toolbar dialog box. A toolbar as shown in Figure W3.7 appears. It has W3 icons and the function of each is labeled in the figure. Six pre‐defined solids like box, sphere, cylinder, cone, wedge and torus can be directly created. Use any one of the following methods to create a solid object: • On the Menu bar click Draw, and from the pull‐down menu select Solids. A sub‐menu appears. Click on the required solid object like box, cylinder, etc. • On the Solid toolbar, click on the desired icon (Figure W3.7). • On the Command line, type a command and press Enter. Predefined solid shapes Setup Box Cone Setup Drawing Sphere Wedge Setup View Cylinder Torus Setup Profile Revolve Section Extrude Slice Interfere Operations to create solids Figure W3.7 A Solids toolbar Page 22 of WEB CHAPTER 3- INTERSECTIONS
W3.6.1 BOX This command is used to create a solid box of a given length , width and height at specified location either by specifying its corner or its center. To draw a box of 100 x 60 x 40 mm size with one corner at the origin the prompt sequence is: 1. Command: BOX ↵ Specify corner of box or [Center] <0,0,0>: ↵ Specify corner or [Cube/Length]: L ↵ Specify L to choose Length option and press Enter. Specify length: 100 ↵ Specify width: 60 ↵ Specify height: 40 ↵ 2. The box created in step 1 does not look 3‐dimensional unless seen in isometric view. To set a viewing direction from a point (1, 1, 1) towards origin use VPOINT command. Command: VPOINT ↵ Specify a viewpoint or [Rotate] <display compass and tripod>: 1, 1, 1 ↵ Specify coordinates of viewing point and press Enter. W3.6.2 SPHERE A SPHERE command draws a solid sphere. The prompt sequence is: Command: SPHERE ↵ Type Sphere and press Enter or click the Sphere icon. Specify center of sphere: Type coordinates of center of sphere and press Enter. Specify radius of sphere or [Diameter]: Type the value of the radius and press Enter. W3.6.3 CYLINDER A solid cylinder with a circular or an elliptical base of the specified size and height can be created using CYLINDER command. If a taper is desired, then a circle can be extruded using EXTRUDE command. The prompt sequence is as under: Command: CYLINDER ↵ Type Cylinder and press Enter or click the icon. Specify center point for base of cylinder or [Elliptical]< 0,0,0>: Specify a center. Specify radius for base of cylinder or [Diameter]: Specify radius and press Enter. Page 23 of WEB CHAPTER 3- INTERSECTIONS
Specify height of cylinder or [Center of other end]: Specify height or center of other end and press Enter key. W3.6.4 CONE This command creates a solid cone, with a circular or an elliptical base. The axis of the cone may not be at right angles to the base. The specified apex point automatically decides its angle for the axis. The prompt sequence below shows the prompts displayed for creating a cone. By default, AutoCAD creates a cone with a circular base by specifying the center and the radius of the base followed by its height. A negative value of the height draws the cone in the negative direction of the Z‐axis. For an elliptical base, type E at the first prompt and then specify the end points of the ellipse and height, etc., as prompted. Command: CONE ↵ Type CONE and press Enter or click the Cone icon. Specify center point for base of cone or [Elliptical]<0,0,0>: Specify coordinates of the center of the base point. Specify Radius for base of cone or [Diameter]: Specify radius R of the base and press Enter. Specify height of cone or [Apex]: Specify height of the cone and press Enter. W3.6.5 WEDGE WEDGE command draws a right‐angled solid wedge with the sloped face tapering along the X‐axis. The following prompts appear with this command: Command: WEDGE ↵ Type Wedge and press Enter or click Wedge icon. Specify first corner of wedge or [Center] <0,0,0>: Specify coordinates of the corner point. Specify corner point or [Cube/Length]: Specify coordinates of a diagonally opposite corner of the base or type C or L to choose Cube/Length options respectively. The Length option prompts are: Specify length: Specify length of the wedge along the X axis. Specify width: Specify width of the wedge and press. Specify height: Specify height of the wedge and press Enter. Page 24 of WEB CHAPTER 3- INTERSECTIONS
W3.6.6 TORUS TORUS command creates a shape similar to an O ring. The required data is the coordinates of the center, radius of the torus from the center to the outside radius and not up to the center line of the tube and the tube radius (Figure W3.8). The prompt sequence is: Command: TORUS ↵ Type Torus and press Enter or click the Torus icon Specify center of torus: Specify coordinates of the center and press Enter Specify radius of torus or [Diameter]: Specify torus outside radius Specify radius of tube or [Diameter]: Specify tube radius Center point Torus diameter Figure W3.8 A torus Tube diameter W3.6.7 EXTRUDE The EXTRUDE command is used to create solids by extruding (adding thickness) to a selected 2‐D object along a path, or by specifying a height value and a tapered angle. First, draw the object and the path. Then use this command. The prompt sequence is as under: Command: EXTRUDE ↵ Select objects: Select the object for extrusion. Select objects: 1 found Specify height of extrusion or [Path]: Specify height or Path. Specify angle of taper for extrusion <0>: If height is specified then only it prompts for taper angle otherwise not. Page 25 of WEB CHAPTER 3- INTERSECTIONS
W3.6.8 REVOLVE Closed objects like polylines, polygons, circles, ellipses, donuts, etc., can be revolved about an axis to create a solid object. Only one object can be revolved at a time. First, draw the object to revolve and the axis about which to revolve. Then use this command. The prompt sequence is given below: Command: REVOLVE ↵ Select objects: Select the object to revolve by clicking over it. Select objects: 1 found Specify start point for axis of revolution or define axis by [Object/X (axis)/Y (axis)]: Click on one end of the axis. Specify endpoint of axis: Click on the other end of the axis. Specify angle of revolution <360>:↵ Press Enter to rotate full revolution. W3.6.9 SLICE This command does not create any new object but is used to slice an existing object in two parts by a slicing plane. One or both the parts can be retained. The slicing plane is defined by many methods suggested in the prompt like object, a plane, Z‐axis and 3 points. The prompt sequence is as under: Command: SLICE ↵ Select objects: Click on the object to be sliced Select objects: 1 found Specify first point on slicing plane by [Object/Zaxis/View/XY/YZ/ZX/3points] <3points>: Click a corner of the plane Specify second point on plane: Click another corner of the plane Specify third point on plane: Click third corner of the plane Specify a point on desired side of the plane or [keep Both sides]: Click on the lower side of the plane to remain lower object. W3.6.10 INTERFERE This command creates a new solid from the common volume of two intersecting solids retaining the original objects. It is similar to the INTERSECT command but that command removes the original objects. The prompt sequence for this command is as under: Page 26 of WEB CHAPTER 3- INTERSECTIONS
Command: INTERFERE ↵ Select first set of solids: Click on one object Select objects: ↵ Press Enter to end first selection. Select second set of solids: Click on the other object Select objects: ↵ Press Enter to end second selection. Create interference solids? [Yes/No] <N>: y↵ Specify the original objects are required or not. Note: The other commands shown on the solids toolbar are beyond the scope of the book. W3.7 COMPOSITE SOLIDS A composite solid is created from two or more solids by any one of the following operations: •
•
•
Union Subtraction Intersection These tools are available on the Solids Editing toolbar. On the Menu bar, click View and then from the pull‐down menu select Toolbars…. From the Toolbars dialog box click on the Solids editing check box. Close the toolbar dialog box. A toolbar as shown in Figure W3.9 appears. Only the first 3 icons are described for the scope of this book. Union Subtract Intersection Figure W3.9 Solids Editing toolbar W3.7.1 JOINING SOLIDS (UNION COMMAND) This command creates a composite region or solid by adding the selected two or more solids. The prompt sequence is as under: Command: UNION ↵ Type Union and press Enter or click the Union icon Select objects: Click on the first solid Select objects: Click on the second solid Select objects: This prompt repeats …………… Select objects: ↵ Press Enter when selection of all objects is finished Page 27 of WEB CHAPTER 3- INTERSECTIONS
W3.7.2 SUBTRACTING SOLIDS (SUBTRACT COMMAND) The SUBTRACT command creates a composite solid by subtracting one solid from the other. A single new composite solid or region is created after subtraction. Command: SUBTRACT ↵ Type Subtract, press Enter or click the Subtract icon Select solids and regions to subtract from… Select objects: Click on the object from which the other object is to be subtracted. Select objects: ↵ Press Enter key when you finish selections. Select solids and regions to subtract … Select objects: Click on the object that is to be subtracted. Select objects: ↵ Press Enter key when you finish selection. W3.7.3 INTERSECT COMMAND The INTERSECT command creates a new solid from the common volume of two or more than two solids and removes the portions outside the common volume. Command: INTERSECT ↵ Select objects: Select objects: …………….. Select objects: ↵ Type Intersect and press Enter or click the icon Select first object. Click on the second object. This prompt repeats. Press Enter key after selecting all objects. EXAMPLE 11 – Creating solids AIM: • To demonstrate the method to create solids at specified locations. • Modify color and shade Creating Solids • Draw a box of 100‐mm length, 80‐mm width and 15‐mm height with one corner at the origin. • Centrally above the box, create a cylinder having a diameter of 60 mm and a height of 30 mm (Figure W3.S11). • At the center of the top of the cylinder, create a right circular cone having a diameter of 40 mm and a height of 40 mm. • Place a solid sphere of 20‐mm diameter at the apex of the cone. • Join all the above solids to create one single solid. Page 28 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.S11 Solid modeling SOLUTION 1. Create a box of length = 100 mm, width = 80 mm and height = 15 mm with one corner at the origin. Command: BOX ↵ Specify corner of box or [Center] <0,0,0>: ↵ Specify corner or [Cube/Length]: L ↵ Specify length: 100 ↵ Specify width: 80 ↵ Specify height: 15 ↵ 2. Create a cylinder of diameter = 60 mm and height = 30 mm above the box. Command: CYLINDER ↵ Specify center point for base of cylinder or [Elliptical]< 0,0,0>: 50,40,15 ↵ Specify radius for base of cylinder or [Diameter]: 30 ↵ Specify height of cylinder or [Center of other end]: 30 ↵ 3. Create a right circular cone of diameter = 40 mm and height = 40 mm above the cylinder. Command: CONE ↵ Specify center point for base of cone or [Elliptical]<0,0,0>: 50,40,45 ↵ Specify Radius for base of cone or [Diameter]: 20 ↵ Specify height of cone or [Apex]: 40 ↵ 4. Create a sphere of diameter = 20 mm at the apex of the cone. Command: SPHERE ↵ Specify center of sphere: 50,40,95 ↵ Specify radius of sphere or [Diameter]: 10 ↵ Page 29 of WEB CHAPTER 3- INTERSECTIONS
5. Viewing in isometric view Command: VPOINT ↵ Specify a viewpoint or [Rotate] <display compass and tripod>: 1, 1, 1 ↵ 6. Notice that a cylinder or sphere, etc., do not look like 3‐D objects. So increase the number of isolines. Command: Isolines ↵ Enter new value for isolines <4>: 32 ↵ 7. Nothing happens on the screen. So regenerate the model using REGEN command. Command: REGEN ↵ Now the view changes and the curved surfaces are shown by 32 lines instead of 4. 8. Joining the solids Command: UNION ↵ Select objects: Select objects: Select objects: Select objects: Select objects: ↵ Type Union and press Enter or click the Union icon. Click on the box. Click on the cylinder. Click on the cone. Click on the sphere. Press Enter when all selections are finished. All the objects are joined. To check this, click anywhere on this model. All the objects are selected. 9. Changing color of the object. Click Modify on Menu bar, and select Properties from the drop‐down list. Select object and then click in the box opposite to color. An arrow appears. Click the arrow and chose Blue color. Close the Properties dialog box. The color of the object now changes to blue. 10. Shading the object At the command line, type SHADE and press Enter. The model is shaded. Page 30 of WEB CHAPTER 3- INTERSECTIONS
EXAMPLE 12 – Intersection of a cone with a plane AIM • To draw a curve of intersection A right circular cone of base radius = 40 mm and height = 100 mm is cut by an inclined rectangular plane of 60 mm x100 mm such that it cuts the base of the cone at 30 mm away from the center line (Figure W3.S12A). The other end of the plane is at a height of 50 mm. (A) A cone and inclined plane (B) Cone cut by inclined plane Figure W3.S12 Intersection of a cone with an inclined plane SOLUTION 1. Draw the cone. Command: CONE ↵ Specify center point for base of cone or [Elliptical]<0,0,0>: 0,0,0 ↵ Specify Radius for base of cone or [Diameter]: 40 ↵ Specify height of cone or [Apex]: 100 ↵ 2. Draw a plane. Calculate the coordinates of each corner of the plane and specify for a 3D face. Specify as under: Command: 3DFACE ↵ Specify first point or [Invisible]: 30,‐30,0 ↵ Specify second point or [Invisible]: 30,30,0 ↵ Specify third point or [Invisible] <exit>: ‐30,30,50 ↵ Specify fourth point or [Invisible] <create three‐sided face>: ‐30,‐30,50 ↵ Specify third point or [Invisible] <exit>: ↵ 3. Change viewing direction using VPOINT command or click SE Isometric button on the View toolbar. Command: VPOINT ↵ Current view direction: VIEWDIR=0.0000,0.0000,1.0000 Specify a view point or [Rotate] <display compass and tripod>: 1,1,1 ↵ Page 31 of WEB CHAPTER 3- INTERSECTIONS
4. Change Isolines from 4 to 32 and regenerate the model. Command: ISOLINES ↵ Enter new value for ISOLINES <4>: 32 ↵ Command: REGEN ↵ A figure as shown in W3.S12A is created. 5. Cut the cone by plane using SLICE command. Command: SLICE Select objects: Click on cone. Select objects: 1 found Specify first point on slicing plane by [Object/Zaxis/View/XY/YZ/ZX/3points] <3points>: Click a corner of the plane. Specify second point on plane: Click another corner of the plane. Specify third point on plane: Click third corner of the plane. Specify a point on desired side of the plane or [keep Both sides]: Click anywhere on lower side of the plane.
A figure as shown in W3.S12B is created. 6. To see the curve of intersection clearly, reset the isolines to 4 as done in the step 4 and regenerate the model. Click on the left‐side view icon on the View toolbar. Click on plane and press the Del key to delete the plane. A view similar to shown in Figure W3.S12C appears. Curve of Intersection
Figure W3.S12(C) Intersection curve of a cone with an inclined plane Page 32 of WEB CHAPTER 3- INTERSECTIONS
EXAMPLE 13 – Intersection of a cylinder and a cone AIM • To draw curve of intersection of two solids Draw a right circular horizontal cone of diameter = 40 mm and height = 80 mm through the center of a vertical cylinder of 30‐mm radius and 90‐mm height such that the center of the cone and the center of the cylinder coincide. (A) Isometric views (B) Side view Figure W3.S13 Intersection curve of a cylinder with a cone SOLUTION 1. Create a cylinder of radius = 30 mm and height = 90 mm. Command: CYLINDER ↵ Specify center point for base of cylinder or [Elliptical]< 0,0,0>: 0,0,0 ↵ Specify radius for base of cylinder or [Diameter]: 30 ↵ Specify height of cylinder or [Center of other end]: 90 ↵ 2. Shift UCS at the center of base of cone Command: Click Origin UCS icon shown above on UCS toolbar. Specify new origin point <0,0,0>: ‐40,0,45↵ Specify coordinates of UCS origin and press Enter. Page 33 of WEB CHAPTER 3- INTERSECTIONS
3. Rotate UCS about Y‐axis Command: Click Y‐axis and rotate icon shown above on UCS toolbar. Specify rotation angle about Y‐axis <90>:↵ Press Enter to accept 90° 4. Create a right circular cone of diameter = 40 mm and height = 40 mm above the cylinder. Command: CONE ↵ Specify center point for base of cone or [Elliptical]<0,0,0>: 50,40,45 ↵ Specify Radius for base of cone or [Diameter]: 20 ↵ Specify height of cone or [Apex]: 40 ↵ EXAMPLE 14 – Intersection of a taper cylinder and a hexagonal inclined prism AIM • To demonstrate use of EXTRUDE and REVOLVE commands to create 3D solids • To demonstrate 3D Boolean operations A taper cylinder has a base diameter of 90 mm, a top diameter of 70 mm and a height of 120 mm( Figure W3.S14A). It has a taper hole with a diameter of 50 mm at the top and 30 mm at the bottom. An inclined tapered hexagonal pyramid having each edge of 20 mm, and height of 160 mm, taper angle of 5° and axis angle of 30° with the horizontal, cuts the cylinder. The center of the base of the prism is 70 mm away from axis of cylinder and at a height of 20 mm. Do the following: (A) Create the solids and join them to get the curve of intersection (UNION command). (B) Subtract the prism from the cylinder (SUBTRACT command). (C) Show the common volume of both the solids (INTERSECT command). Figure W3.S14 (A) ‐ A hollow cylinder and a pyramid in front view Page 34 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.S14 (B) A hollow cylinder and a pyramid in pictorial view SOLUTION A. CREATING SOLIDS A1. Draw the taper hollow cylinder using REVOLVE command. 1. Draw a vertical line of 120‐mm length using LINE command as an axis of the cylinder. 2. Use OFFSET command with a distance of 15 mm to draw another parallel line to previous line. 3. Use PLINE command to draw a trapezium of the sizes as given below. It may be noted that objects created by LINE command cannot be revolved. Command: PLINE ↵ Specify start point: Click at the bottom of the offset line. Specify next point or [Arc/Halfwidth/Length/Undo/Width]: @30,0 ↵ Specify next point or [Arc/Close/Halfwidth/Length/Undo/Width]: @‐10,120 ↵ Specify next point or [Arc/Close/Halfwidth/Length/Undo/Width]: @‐10,0 ↵ Specify next point or [Arc/Close/Halfwidth/Length/Undo/Width]: C ↵ 4. Delete the line created in the step 2. 5. Use REVOLVE command to create the cylinder. Command: REVOLVE ↵ Select objects: Click on the trapezium created in Step 3. Specify start point for axis of revolution or define axis by [Object/X (axis)/Y (axis)]: Click at bottom of the vertical line drawn in Step 1. Page 35 of WEB CHAPTER 3- INTERSECTIONS
Specify endpoint of axis: Click at top of the vertical line drawn inSstep 1. Specify angle of revolution <360>: ↵ Press Enter to revolve for full revolution. 6. Set isolines as 32 as explained in previous examples and regenerate the view. A2. Draw a hexagonal inclined pyramid using EXTRUDE command. 7. Draw a line using LINE command with start point as (‐70, 20, 0) and next point @ 160 < 30. This line will be used as an axis of the pyramid. First, set UCS so that the X‐Y plane is inclined at 30° and the Z0axis is along the axis of the inclined pyramid. To do this, follow the steps 8, 9 and 10 given below: 8. Click Origin UCS icon on UCS toolbar and specify origin as (‐70, 20, 0). It is the center of the base of the pyramid. 9. Click Z‐axis rotate icon on UCS toolbar and specify the angle of rotation as 30°. 10. Click Y‐axis rotate icon on UCS toolbar and specify the angle of rotation as 90°. 11. Draw the base of the pyramid using POLYGON command as under: Command: POLYGON ↵ Enter number of sides <4>: 6 ↵ Specify center of polygon or [Edge]: Click on UCS new origin Enter an option [Inscribed in circle/Circumscribed about circle] <I>:↵ Press Enter Specify radius of circle: 20 ↵ Specify radius and it is the edge distance as well. 12. Create prism by EXTRUDE command as follows: Command: EXTRUDE ↵ Select objects: Click on hexagon Specify height of extrusion or [Path]: 160 ↵ Specify height as 160 and press Enter. Specify angle of taper for extrusion <0>: 5 ↵ Specify taper angle 5° and press Enter. B. Joining the two solids to get the curve of intersection 13. Joining solids Command: UNION ↵ Type Union and press Enter or click the Union icon. Select objects: Click on the cylinder. Select objects: Click on the pyramid. Select objects: ↵ Press Enter. See the curve of intersection at the common boundary of both the solids in Figure W3.S14B. It is created automatically. Page 36 of WEB CHAPTER 3- INTERSECTIONS
14. Subtracting solids Command: SUBTRACT ↵ Type Subtract, press Enter or click the Subtract icon Select objects: Click on the cylinder. Select objects: ↵ Press Enter. Select objects: Click on the pyramid. Select objects: ↵ Press Enter. A tapered hexagonal hole is created in the cylinder as shown in Figure W3.S14C. 15. Finding common volume of solids Command: INTERSECT ↵ Type Intersect and press Enter or click the icon Select objects: Click on the cylinder. Select objects: Click on the pyramid. Select objects: ↵ Press Enter. The volume common to both the solids is created as shown in Figure W3.S14D. Curve of
Intersection
Figure W3.S14(B) Figure W3.14(C) Figure W3.S14(D) Figure S14‐ Use of UNION, SUBTRACT and INTERSECT commands Page 37 of WEB CHAPTER 3- INTERSECTIONS
THEORY QUESTIONS Q 1. What is meant by curve of intersection? Explain by giving a neat sketch. Q 2. Describe a method to draw the curve of intersection when a cylinder is cut by an inclined plane. Q 3. Explain the method to draw curve of intersection when a cone is cut by a cylinder. Q 4. Give any two examples of two intersecting objects being used practically. CAD Q 5. Differentiate between surface and solid model. Q 6. What are the options offered in drawing a solid box? Q 7. Write the command sequence for drawing a solid CYLINDER. Q 8. How will you draw a cone with an elliptical base? Q 9. Sketch a torus and show the tube and torus radii. Q 10. What type of operations can be done using Solid Editing commands? Q 11. Describe the different icons on the Solids Editing toolbar. Q 12. Explain the use of SOLIDEDIT command. Q 13. Explain the UNION command in detail by giving sketches. Q W3. Differentiate between INTERSECT and SUBTRACT commands. FILL UP THE BLANK QUESTIONS Fill up the blanks by appropriate word(s) 1. When two solid objects cut each other, their common boundary is called the curve of ___________. 2. If a cylinder is cut by an inclined plane, the curve of intersection is ____________. 3. The curve of intersection of two axially intersecting pipes of equal diameter is ____________. CAD 4. A solid is drawn tapered along _____________ axis. 5. The viewing direction can be changed by _____________ command. 6. The lines showing a curved solid area are called _____________. 7. The system variable to set the number of isolines is _____________. 8. A taper cylinder can be drawn by _____________ command. 9. A cone drawn by CONE command can have circular or _____________ base. 10. Torus radius is the radius from center point to _____________. 11. A 2‐D object to make 3‐D can be moved along a specified path by _____________ command. 12. Axis symmetric solids can be created by _____________command 13.
SLICE command _____________ an object and makes in _____________parts. 14. A new solid is created from the common volume of two solids by ______ and _____ commands. 15. INTERFERE command _____________ the original objects. 16. Curve of intersection is _____________ created in intersecting 3D solids. Page 38 of WEB CHAPTER 3- INTERSECTIONS
17. The command to joint two or more solids is _____________. 18. The command to remove common volume of two solid objects is_____________. MULTIPLE CHOICE QUESTIONS Q 1. A curve of intersection is seen when two objects are a. placed side by side b. placed at right angles to each other c. intersect each other at any angle d. intersect each other at right angles Q 2. Intersections are shown to give a. more details about the shape of an object b. actual shape of intersecting surfaces c. internal details of object d. to hide the actual shape Q 3. To draw curve of intersection of two objects a. a minimum two orthographic views are required b. all three views are necessary c. auxiliary view is needed d. sectional view is required Q 4. Shape of curve of intersection is a. always a curve b. depends upon shape and orientation of objects c. a straight line d. an arc Q 5. Two pipes of equal diameter meet each other axially at right angles. The curve of intersection will be a. an arc b. a general curve c. a straight line d. an ellipse Page 39 of WEB CHAPTER 3- INTERSECTIONS
CAD Q6. Specifying some parameters can make a solid box. Which of the following is an invalid set? a. Center point of box, length, width and height b. 2 diagonal corners of box c. Length, width and height d. Center point of box and cube length Q7. CYLINDER command is used to draw a. only a solid circular cylinder b. a solid cylinder with an elliptical or circular base c. only a solid cylinder with a circular base d. a solid or a hollow cylinder with a circular base Q8. A solid cone with an elliptical base is drawn using a. SOLID CONE command b. ELLIPSE and CONE command c. REVSURF and CONE command d. CONE command Q 9. A SPHERE command is used to draw a a. spherical surface b. hollow sphere c. solid sphere d. any one of the above Q 10. A Slanting face of a solid WEDGE is along the a. X‐axis b. Y‐axis c. Z‐axis d. X or Y or Z axis Q 11. Torus radius is defined as the distance from its center to a. outer periphery of tube b. center line of the tube c. inner periphery of tube d. inner periphery of tube/2 Q 12. Isolines can be set for any value between a.
b.
c.
d.
1 to 16 4 to 255 4 to 2047 16 to 4095 Page 40 of WEB CHAPTER 3- INTERSECTIONS
Q 13. UNION command is used to join a. only 2 D objects b. two 3D objects c. two or more than two 3D objects d. two or more than two 2D or 3D objects Q 14. SUBTRACT command is used to a. do calculations while drawing b. subtract areas of two objects c. remove an object from the selection set d. subtract a 3D object from another 3D object Q 15. INTERSECT command gives a a. common point of intersection of two drawing entities b. new object from the common volume of intersecting solids c. common volume of two 3D objects d. point of intersection of two lines only Q 16. The Path for EXTRUDE command has to be a. only a straight line perpendicular to a 2D object b. only a straight line at any angle to a 2D object c. a line or curve perpendicular to a 2D object d. a line or curve at any angle to a 2D object Q 17. EXTRUDE command can extrude objects with a. zero taper b. only positive taper c. only negative positive taper d. positive or negative taper Q 18. REVOLVE command is suitable for objects which are a. cylindrical b. spherical c. cylindrical or conical d. flat Q 19. REVOLVE command can rotate profiles along a. any axis b. X‐axis only c. Y‐axis only d. Z‐axis only Page 41 of WEB CHAPTER 3- INTERSECTIONS
Q 20. Solid objects can be cut to two equal or unequal pieces using the command a. SECTION b. SLICE c. INTERFERE d. EXTRUDE Q 21. The SLICE command divides an object into a.
b.
c.
d.
two equal pieces two equal or unequal pieces more than 2 pieces many pieces of small thickness Q 22. The INTERFERE command a. displaces the overlapping solids b. makes a new solid by combining overlapping solids c. makes a new solid from the common volume of overlapping solids d. deletes the overlapping volume of solids Answers to Fill up the blank questions 1. intersection 2. ellipse 3. straight line CAD 4. X 5.VPOINT 6.I solines 7. Isolines 8. EXTRUDE 9. elliptical 10. outside diameter 11. EXTRUDE 12. REVOLVE 13. Cuts, two 14. INTERSECT, INTERFERE 15. retains 16. automatically 17. UNION 18. SUBTRACT Answers to multiple choice questions 1‐c 2‐b 3‐a 4‐d 5‐c 6‐c 7‐b 8‐d 9‐c 10‐a 11‐a 12‐b 13‐c 14‐d 15‐b 16‐d 17‐d 18‐c 19‐a 20‐b 21‐b 22‐c Page 42 of WEB CHAPTER 3- INTERSECTIONS
ASSIGNMENT ON INTERSECTIONS Draw the orthographic views and the curves of intersection for the following intersecting objects: Q 1. A right circular cone with base of 60 mm and height of 100 mm is cut by a cylinder of 200‐mm diameter at its mid‐height at right angles to its axis. Q 2. A vertical cylinder of 60‐mm diameter and 90‐mm length is intersected by a rod of 25‐mm diameter at its centre such that it forms an angle 60° with the axis of the cylinder and the free end extends 70 mm from center of cylinder.. Q 3. A hexagonal prism of 100‐mm length and each edge of 40 mm is cut by a horizontal triangular prism of each edge of 40 mm and a length of 100 mm at its mid‐height. The vertical face of the triangular prism is 17 mm away from the vertical axis of the hexagonal prism. Q 4. A square pyramid turned at 45° about vertical axis has base of 80 mm and an apex height of 150 mm. It is joined with a round bar of 25‐mm diameter inclined at 45° in the vertical plane at a height of 70 mm and its axis meets the pyramid axis at a height of 70 mm from the base. Q 5. A hexagonal prism of each edge of 40 mm and a height of 130 mm is cut at the middle of its axis by an inclined cylinder having a diameter of 40 mm and a length 160 mm at an angle of 30° with the horizontal. Page 43 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.P1 Assignment on Intersections CAD ASSIGNMENT ON INTERSECTIONS Draw the solid models of the following objects and get the curves of intersection by joining them: Q 6. A cylinder having a diameter of 80 mm and a height of 120 mm is cut by an inclined plane at 45° passing through its center Q 7. A cylinder having a diameter of 100 mm and a height of 150 mm is cut by a horizontal cone at its central axis. The cone has a base of 60 mm and a length 150 mm. The cone protrudes the cylinder equally on both the sides. Q 8. Two pipes having outside diameters of 75 mm and 50 mm are joined to form a T joint. The small pipe is centrally at right angles to the big pipe which has a length of 100 mm and extends 500 mm above. Q 9. A triangular prism has one rectangular face in the vertical plane, each edge of 70 mm and a height of 90 mm. It is cut centrally by a square prism having each edge of 30 mm and a length of 70 mm turned by 45° about horizontal axis. The square prism protrudes the triangular prism equally on both the sides. Page 44 of WEB CHAPTER 3- INTERSECTIONS
Figure W3.P2 CAD Assignment on Intersections Page 45 of WEB CHAPTER 3- INTERSECTIONS
PROBLEMS FOR PRACTICE ON INTERSECTION Draw the orthographic views and the curves of intersection for the following intersecting objects: Q 10. A square prism having a base of 25 mm x 25 mm and a height of 60 mm is turned at 30° about its vertical axis. It is cut by another rectangular prism having a base of 20 mm x 15 mm and a length of 80 mm such that its longer side is inclined at 45° with the horizontal plane at a height of 20 mm from the base of the square prism. Q 11. A pentagonal prism having each edge of 30 mm and a height of 120 mm has its one rectangular face in the principle planes. A cylinder having a diameter of 30 mm, a length of 150 mm inclined at 30° cuts the prism. Axes of both the objects meet in the center at the mid‐height of the prism. Q 12. A cone having a base diameter of 80 mm and an apex height of 120 mm is cut by an inclined cylinder having a diameter of 36 mm and a length of 120 mm at angle of 30° centrally at the axis of the cone. Q 13. Both the solids have same sizes as given in Q 12 but the cylinder is offset by 7 mm from the axis of the cone in the horizontal plane. Q 14. A vertical pipe having a diameter of 80 mm and a length of 160 mm is intersected by another inclined pipe having a diameter of 35 mm and a length of 100 mm. Their centers are offset by 10 mm in the horizontal plane. The small pipe protrudes equally on both the sides. Page 46 of WEB CHAPTER 3- INTERSECTIONS
PROBLEMS FOR PRACTICE ON SOLID MODELING Set the isolines as 32 and viewpoint as (1, ‐1, 1) and then draw the following solid objects: Q 15. A solid box with one corner at (125, 35, 0) and the diagonal corner at (175, 105, 45). Q 16. A solid box with one corner at (0, 0, 0) and length =115 mm, width = 80 mm and height = 50 mm. Q 17. A solid cube with each side = 60 mm. It can be located anywhere on the screen. Q 18. A solid box with its center at (250, 15, 20), length = 100 mm, width = 30 mm and height = 40 mm. Q 19. A solid sphere has a radius of 15 mm such that it rests centrally on the top of box drawn in Question 15. Q 20. A solid cone with the base anywhere having a diameter of 40 mm and a height of 60 mm. Q 21. A solid cylinder with its base at the origin, and a radius of 30 mm such that the top center is at (0, 20, 70). Q 22. A solid wedge with the length along the slanting face = 80 mm, width = 60 mm and height = 60 mm. Position it, in such a manner that its vertical face touches any face of the cube drawn in Q. 17. Q 23. A torus with a radius of 80 mm, tube diameter of 50 mm and center at the top of the apex of the cone in Q 20.