MATHEMATICS FOR SYSTEMATIC MODELING_3
ROTATION AND CIRCLE
ROTATION BY CENTER + AXIS + ANGLE
ROTATION BY CENTER + PLANE + ANGLE
P : plane
v : axis
p : center
x’ : result
n : plane normal
x’ : result
p : center
A : angle
A : angle
x : target
x : target
PROJECTION VS ROTATION
(1) Projection on XY plane
(2) Rotate around X-axis
(3) Rotate around axis on XY-plane
perpendicular to a vector to the point
(4) Rotate around arbitrary axis on
XY-plane
(4)
(1) (2) (3)
PROJECTION / EXTENSION / ROTATION
How to move corner point to be on XY-plane?
ROTATION
EXTENSION
PROJECTION
h
h
h
w
w’’
w’
ANGLE OF PLANES
ANGLE OF SURFACES
(1) Intersectional line
of two planes
(2) Line on each plane
perpendicular to the
intersectional line
(3) Angle is measured
by the perpendicular
lines
(1) Tangent line at the
measurement point
(2) Perpendicular
plane to the tangent
line
(3) Intersection curve
of the plane with the
surfaces
(4) Tangent lines to the
intersectional curve
(5) Angle is measured
by the tangent lines
(2)
(3)
(2)
(1)
(4)
(3)
(5)
(3)
(4)
(2)
(1)
MATHEMATICS FOR SYSTEMATIC MODELING_3
ROTATION AND CIRCLE
DEFINITION OF CIRCLE
CENTER + AXIS + RADIUS
CENTER + PLANE + RADIUS
PLANE (OR AXIS) + DIAMETER
P : plane
v : axis
v : axis
p : center
p : center
p2 : 2nd point of diameter
P : plane
r : radius
r : radius
2 TANGENT LINES + RADIUS
3 POINTS
p2 : point on circle
r : radius
L2 :tangent line
p1 : 1st point of diameter
3 TANGENT LINES
bisector
bisector
bisector
L2 :tangent line
p1 : point on circle
L3 :
tangent line
g
(3)
d
(3)
f
e
(1)
(1) Draw two lines
(2) Draw perpendicular line
on midpoint
(3) Take intersection
(1) Draw two parallel lines
(2) Connect midpoints and extend
until intersecting with ellipse
(3) Take midpoint
h
(3)
h
h
(2)
d
center
b
(2)
f
e
(3)
center
(2)
(2)
a
b
(1)
a
(1)
(1)
g
(2)
bisector
(1)
(2)
g
FINDING CENTER
L1 :tangent line
h (2)
g
L1 :tangent line
m
axiinor
s
p3 : point on circle
m
axiajor
s
(1) Draw a circle at the ellipse center
(2) Take intersection and connect them
to form a rectangle
(3) Connect each midpoint of edges
DIVIDING ARC
EQUAL EDGE LENGTHS
UNIQUE END ANGLES
EQUAL ANGLES
UNIQUE END EDGE LENGTHS
EQUAL ANGLES
EQUAL EDGE LENGTHS
MATHEMATICS FOR SYSTEMATIC MODELING_3
ROTATION AND CIRCLE
INTERSECTION OF PLANE AND CYLINDER OR CONE
CIRCLE
ELLIPSE
CIRCLE
ELLIPSE
PARABOLA
LOFTED SURFACE BETWEEN CIRCLES VS TILTED CYLINDER AND CONE
CYLINDER
LOFTED
TILTED
CONE LOFTED
TILTED
PANELIZATION OF CYLINDER AND CONE
CYLINDER
SHEARED CYLINDER
CONE
SHEARED CONE
TILTED
CYLINDER
TILTED CONE
TILTED
CYLINDER
TILTED CONE
HYPERBOLA
MATHEMATICS FOR SYSTEMATIC MODELING_3
ROTATION AND CIRCLE
AFFINE TRANSFORMATION
() () ()
x’
y’
=
x
y
+
tx
ty
TRANSLATION
( ) ( )( )
( ) ( )( )
NON-UNIFORM SCALING
UNIFORM SCALING
( ) ( )( )
() (
x’
y’
x’
y’
=
=
sx 0
0 sy
rx 0
0 ry
x
y
x’
y’
x’
y’
x
y
rx, ry = -1 or 1
REFLECTION
or
=
=
s 0
0 s
cos
sin
ROTATION
( ) ( )( )
( ) ( )( )
x’
y’
=
1 k
0 1
x
y
(on x-axis)
x’
y’
=
1 0
k 1
x
y
(on y-axis)
SHEARING
EXERCISE
TASK1. BASIC
TASK2. ADVANCED
Draw corner with an arbitrary
angle and draw facet lines
whose angles are all equal.
Measure the angle to make
sure they are equal.
Model tilted faces with
the facet lines drawn
in task 1 whose unfolded
shapes are identical
and the angles of faces
are all equal.
Measure the angles of
faces to check if they
are equal and unfold
the faces to check if
the shapes are identical.
* The angle of tilted
faces in 3D is different
from the angle on the
plan.
x
y
-sin
cos
)( )
x
y