Curves and Surfaces
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Representation of
Curves & Surfaces
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Polygon Meshes
Parametric Cubic Curves
Parametric Bi-Cubic Surfaces
Quadric Surfaces
Specialized Modeling Techniques
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The Teapot
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Representing Polygon Meshes
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explicit representation
by a list of vertex coordinates
pointers to a vertex list
pointers to an edge list
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Pointers to a Vertex List
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Pointers to an Edge List
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Plane Equation
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Ax+By+Cz+D=0
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And (A, B, C) means the normal vector
so, given points P1, P2, and P3 on the plane
(A, B, C) =P1P2  P1P3
What happened if (A, B, C) =(0, 0, 0)?
The distance from a vertex (x, y, z) to the plane is
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Parametric Cubic Curves
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The cubic polynomials that define a
curve segment
are of the form
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Parametric Cubic Curves
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The curve segment can be rewrite as
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where
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Continuity between
curve segments
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Tangent Vector
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Continuity between
curve segments
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G0 geometric continuity
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two curve segments join together
G1 geometric continuity
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the directions (but not necessarily
the magnitudes) of the two
segments’ tangent vectors are equal
at a join point
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Continuity between
curve segments
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C1 continuous
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the tangent vectors of the two cubic
curve segments are equal (both
directions and magnitudes) at the
segments’ join point
Cn continuous
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the direction and magnitude of
through the nth derivative are equal
at the join point
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Continuity between
curve segments
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Continuity between
curve segments
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Three Types of
Parametric Cubic Curves
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Hermite Curves
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Bézier Curves
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defined by two endpoints and two
endpoint tangent vectors
defined by two endpoints and two
control points which control the
endpoint’ tangent vectors
Splines
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defined by four control points
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Parametric Cubic Curves
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Rewrite the coefficient matrix as
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where M is a 44 basis matrix, G is
called the geometry matrix
so
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Parametric Cubic Curves
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where
function
is called the blending
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Hermite Curves
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Given the endpoints P1 and P4 and tangent
vectors at R1 and R4
What are
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Hermite basis matrix MH
Hermite geometry vector GH
Hermite blending functions BH
By definition
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Hermite Curves
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Since
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Hermite Curves
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so
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and
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Bezier Curves
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Four control points
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Two endpoints, two direction points
Length of lines from each endpoint to its direction
point representing the speed with which the curve
sets off towards the direction point
Fig. 4.8, 4.9
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Bézier Curves
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Given the endpoints and and two
control points and which determine
the endpoints’ tangent vectors, such
that
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What is
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Bézier basis matrix MB
Bézier geometry vector GB
Bézier blending functions BB
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Bézier Curves
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by definition
then
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so
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Bézier Curves
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and
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Subdividing Bézier Curves
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How to draw the curve ?
How to convert it to be linesegments ?
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Bezier Curves
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Constructing a Bezier curve
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Fig. 4.10-13
Finding mid-points of lines
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Bezier Curves
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Convex Hull
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Spline
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the polynomial coefficients for natural
cubic splines are dependent on all n
control points
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has one more degree of continuity than
is inherent in the Hermite and Bézier
forms
moving any one control point affects the
entire curve
the computation time needed to invert
the matrix can interfere with rapid
interactive reshaping of a curve
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B-Spline
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Uniform NonRational B-Splines
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cubic B-Spline
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uniform
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has m+1 control points
has m-2 cubic polynomial curve
segments
the knots are spaced at equal
intervals of the parameter t
non-rational
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not rational cubic polynomial curves
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Uniform NonRational B-Splines
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Curve segment Qi is defined by points
thus
B-Spline geometry matrix
if
then
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Uniform NonRational B-Splines
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so B-Spline basis matrix
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B-Spline blending functions
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NonUniform NonRational
B-Splines
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the knot-value sequence is a
nondecreasing sequence
allow multiple knot and the
number of identical parameter is the
multiplicity
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Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5)
so
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NonUniform NonRational
B-Splines
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Where
is the jth-order blending
function for weighting control point pi
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Knot Multiplicity &
Continuity
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Since Q(ti) is within the convex hull of
Pi-3, Pi-2, and Pi-1
If ti=ti+1, Q(ti) is within the convex hull of
Pi-3, Pi-2, and Pi-1 and the convex hull of Pi-2,
Pi-1, and Pi, so it will lie on Pi-2Pi-1
If ti=ti+1=ti+2, Q(ti) will lie on pi-1
If ti=ti+1=ti+2=ti+3, Q(ti) will lie on both Pi-1
and Pi, and the curve becomes broken
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Knot Multiplicity &
Continuity
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multiplicity
multiplicity
multiplicity
multiplicity
1
2
3
4
:
:
:
:
C2 continuity
C1 continuity
C0 continuity
no continuity
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NURBS:
NonUniform Rational B-Splines
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rational
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x(t), y(t) and z(t) are defined as the ratio of two
cubic polynomials
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rational cubic polynomial curve segments are
ratios of polynomials
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can be Bézier, Hermite, or B-Splines
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Parametric Bi-Cubic Surfaces
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Parametric cubic curves are
so parametric bi-cubic surfaces are
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If we allow the points in G to vary in 3D
along some path, then
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since Gi(t) are cubics
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Parametric Bi-Cubic Surfaces
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so
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Hermite Surfaces
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Bézier Surfaces
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B-Spline Surfaces
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Normals to Surfaces
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Quadric Surfaces
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implicit surface equation
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an alternative representation
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Quadric Surfaces
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advantages
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computing the surface normal
testing whether a point is on the
surface
computing z given x and y
calculating intersections of one
surface with another
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