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Curves and Surfaces

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Representation of Curves & Surfaces

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 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 Ax+By+Cz+D=0

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 The cubic polynomials that define a

curve segment are of the form

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Parametric Cubic Curves The curve segment can be rewrite as

where

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Continuity between curve segments

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Tangent Vector

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Continuity between curve segments

G0 geometric continuity two curve segments join together

G1 geometric continuity 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

C1 continuous the tangent vectors of the two cubic curve

segments are equal (both directions and magnitudes) at the segments’ join point

Cn continuous 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

Hermite Curves defined by two endpoints and two endpoin

t tangent vectors Bézier Curves

defined by two endpoints and two control points which control the endpoint’ tangent vectors

Splines defined by four control points

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Parametric Cubic Curves Rewrite the coefficient matrix as

where M is a 44 basis matrix, G iscalled the geometry matrix

so

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Parametric Cubic Curves

where is called the blending function

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Hermite Curves Given the endpoints P1 and P4 and tangent

vectors at R1 and R4

What are Hermite basis matrix MH

Hermite geometry vector GH

Hermite blending functions BH

By definition

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Hermite Curves Since

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Hermite Curves so

and

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Bezier Curves Four control points

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 Given the endpoints and and two contr

ol points and which determine the endpoints’ tangent vectors, such that

What is Bézier basis matrix MB

Bézier geometry vector GB

Bézier blending functions BB

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Bézier Curves by definition then

so

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Bézier Curves and

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Subdividing Bézier Curves How to draw the curve ? How to convert it to be line-

segments ?

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Bezier Curves Constructing a Bezier curve

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 the polynomial coefficients for natural cu

bic splines are dependent on all n control points has one more degree of continuity than is in

herent in the Hermite and Bézier forms moving any one control point affects the ent

ire 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 cubic B-Spline

has m+1 control points has m-2 cubic polynomial curve segment

s uniform

the knots are spaced at equal intervals of the parameter t

non-rational not rational cubic polynomial curves

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Uniform NonRational B-Splines Curve segment Qi is defined by points

thus B-Spline geometry matrix

if then

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Uniform NonRational B-Splines so B-Spline basis matrix

B-Spline blending functions

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NonUniform NonRational B-Splines

the knot-value sequence is a nondecreasing sequence

allow multiple knot and the number of identical parameter is the multiplicity Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5)

so

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NonUniform NonRationalB-Splines

Where is the jth-order blendingfunction for weighting control point pi

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Knot Multiplicity & Continuity

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

multiplicity 1 : C2 continuity multiplicity 2 : C1 continuity multiplicity 3 : C0 continuity multiplicity 4 : no continuity

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NURBS:NonUniform Rational B-Splines

rational x(t), y(t) and z(t) are defined as the ratio of two cu

bic polynomials rational cubic polynomial curve segments are r

atios of polynomials

can be Bézier, Hermite, or B-Splines

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Parametric Bi-Cubic Surfaces Parametric cubic curves are so parametric bi-cubic surfaces are

If we allow the points in G to vary in 3D along some path, then

since Gi(t) are cubics

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Parametric Bi-Cubic Surfaces 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 implicit surface equation

an alternative representation

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Quadric Surfaces advantages

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