Curves and Surfaces

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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

• AxByCzD0
• 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 endpoint 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 4?4 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 control 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 cubic
  splines are dependent on all n control points
• 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

• cubic B-Spline
• has m1 control points
• has m-2 cubic polynomial curve segments
• 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 titi1, 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 titi1ti2, Q(ti) will lie on pi-1
• If titi1ti2ti3, 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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NURBSNonUniform Rational B-Splines

• rational
• x(t), y(t) and z(t) are defined as the ratio of
  two cubic polynomials
• rational cubic polynomial curve segments are
  ratios 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