Advanced Computer Graphics: Parametric Curves and Surfaces

1
Advanced Computer GraphicsParametric Curves and
Surfaces

• James Gain
• Department of Computer ScienceUniversity of Cape
  Town
• jgain_at_cs.uct.ac.za

2
Objectives

• To introduce general concepts associated with all
  parametric curves
• To list the desirable properties of a curve
  formulation
• To consider the smoothness (continuity) and
  rendering of curves
• To derive the Hermite and Power curves

3
Motivation

• Representing curves as polylines, and surfaces as
  a collection of polygons is problematic
• requires large amounts of co-ordinate and
  connectivity information
• difficult to exactly design smoothly varying
  shapes
• Need a compact and easily manipulated high-level
  representation of smoothly varying curves and
  surfaces.
• But must be internally convertible to line
  segment or polygon approximation for polygon-scan
  conversion rendering.

4
Spline

• A thin flexible wooden slat.
• Weights (ducks) fix the spline to go through
  certain points.
• Stiffness makes the curve smooth but not too
  wiggly.
• Features
• Controllable
• Smooth but minimum curvature
• Þ Can be implemented as parametric polynomial
  piecewise curves.

5
Formulations

• Explicit
• y and z represented as functions of x
• 2D line
• Implicit
• function relating all coordinates
• 2D line
• Weaknesses
• may have too many or too few solutions
• affine transformations can be difficult
• vertical tangents (with infinite slope) present
  problems
• difficult to join separate curves together
  smoothly
• Strengths
• good for point on curve tests

6
Parametric Formulation

• Express points on the curve in terms of an
  indirect control parameter, t
• Generally, the parameter is clamped,
• 2D Line
• Weaknesses
• point on curve tests are difficult
• Strengths
• calculating slope
• generating points on the curve
• joining separate curves together
• easier to apply affine transformations

7
Polynomials

• An n-th degree polynomial has the form
• Where are the coefficients and are
  the power terms.
• The degree, , is the highest power to which
  is raised.
• The order is the number of coefficients (
  ).
• Constant
• Linear
• Quadratic
• Cubic

8
Polynomial Curves

• Polynomial curves are represented by a sum of
  polynomials (basis or blending functions) each
  weighted by a 2D or 3D coefficient (control
  points) where the parameter has a restricted
  range
• Often represented in the form
• Where is a point on the
  curve, are the control points
  and are the basis
  functions.
• Example

9
Basis Functions

• The degree of the curve is the degree of the
  highest basis function (sum of polynomials is a
  polynomial)
• There are ordern1 basis functions and control
  points. For instance, a cubic curve has 4 control
  points.
• The contribution of a control point to the
  curve at position is determined (weighted) by
  the basis function .
• We would like there to be an intuitive
  relationship between the control points and the
  curve. This depends on the choice of basis
  functions.
• If successive control points are joined by line
  segments this is called the control polygon.

10
Piecewise Curves

• How do we create interesting and complex curves
• Degree raising
• Increase the degree of the curve (allows more
  inflections).
• Problems cost of operations on the curve depend
  on the degree each control point influences the
  whole of the curve
• Piecewise
• Join the heads and tails of different curves.
• Have to make sure that the joins are smooth.

11
Continuity

• Geometric continuity ( ) is visually
  smooth. The derivative vectors agree in direction
  but not magnitude.
• Full continuity ( ) is parametrically
  smooth. The derivative vectors agree in both
  direction and magnitude.

12
Choosing degree and continuity

• Cubic Degree
• High enough degree to be truly 3D and have an
  inflection point
• Low enough degree to be efficient
• Continuity for Modelling
• Visual appearance is important so geometric
  continuity generally sufficient
• Continuity for Animation
• A camera or object follows the curve usually with
  constant size steps in .
• The change in speed and/or acceleration along the
  curve is important so parametric continuity is
  required.

13
Exercise Properties

• Question
• What features are important for a curve
  representation?
• Answers
• Can guarantee and control levels of continuity
• Compact to store
• Efficient to render and interrogate
• A geometrically intuitive relationship between
  the control points and the curve
• Must not oscillate
• Should be able to represent all shapes
• Easy to transform (scale, rotate, translate) and
  modify

14
Properties

• Endpoint Interpolation
• The curve passes through (interpolates) some (or
  all) of the control points.
• Affine Invariance
• Instead of transforming potentially many
  individual points on the curve, the control
  points are transformed. This is similar to the
  property that affine transformations preserve
  straight lines.

15
More Properties

• Convex Hull Property
• The curve never wanders outside the convex hull.
  This hull is defined as the set of all convex
  combinations of control points
• Variation Diminishing Property
• Curves does not oscillate more than its control
  polygon. In 2D, no straight line can intersect
  the curve more times than it intersects the
  control polygon.

16
Rendering by Evaluation

• Task convert curve to a sequence of line
  segments
• Evaluate the curve at intervals. Join the
  evaluated points.
• Problems length of line segments depends on the
  parametrisation (will be shorter where the
  magnitude of the derivative is smaller) not the
  curvature. No guarantee that the curve will
  appear smooth.

17
Rendering by Subdivision

• Recursively divide the curve in half. If
  sufficiently flat stop recursion, otherwise
  subdivide further.
• Need a means of subdividing and testing
  flatness. Good trade-off between visual quality
  and number of line segments.

18
Power Curves
Power Basis

• Problem very difficult to predict the shape of
  the curve from the position of the control
  points.
• Solution use other basis functions (Hermite,
  Bézier).

19
Hermite Curves

• Intuitive control two endpoints and end tangent
  vectors
• Hermite curve equation

20
Derivation

• Each basis function has four unknowns
• Basis functions must satisfy the following
  properties
• Use this information to solve for the
  coefficients of the basis functions.

21
Formula

• Cubic Hermite Basis

22
Changing length of V0
V0
P1
P0
V1
23
Changing direction of V0
24
Properties

• Limited to cubic curves
• Curve does not lie within the convex hull of its
  control points
• C1 continuity is very easy to enforce (start and
  end points and vectors are set equal)
• More popular for animation than modelling

25
Chaikins Corner Cutting

• Take a control polygon and cut each edge at the
  and marks.
• Join these new points across the corners to
  create a refined (subdivided) control polygon.
• This process can be repeated, producing a
  successively smoother curve.
• In the limit, a quadratic curve.
• Note each control point only influences a local
  portion of the curve.

26
Objectives

• To explore Bézier curves in detail
• To briefly motivate for B-spline curves
• To compare and contrast the curve formulations
  described so far
• To extend from curves to surfaces
• To describe the shortcomings of standard surfaces
  and show how subdivision surfaces avoid them

27
Bézier Curves

• Discovered independently in the early 70s by
  Bézier and de Casteljau. Both worked in the
  automobile industry.
• Bézier curves are popular because there is a
  clear geometric connection between the control
  points and the curve
• Interpolates the first and last control points
  
• The tangent vectors at the endpoints are aligned
  with the line segments, and
  
• The curve lies in the convex hull of its control
  points. This is a key difference from Hermite
  curves.

28
Bézier (Bernstein) Basis

• Binomial coefficient function
• Read as choose , it is the number of
  ways of choosing items from a collection of
  items.
• The binomial coefficients can be quickly derived
  using Pascals triangle (terms are summed from
  above diagonally left and right)

29
Linear and Quadratic Bézier Curves

• Linear (a line segment)
• Quadratic

30
Cubic Bézier Curves

• Cubic

31
Examples
32
Properties

• Endpoint interpolation
• Convex Hull
• Curve is an affine combination of the control
  points and so lies inside the convex hull.
• Affine Invariance
• An affine transformation of the control points is
  the same as transforming the curve (not true of
  perspective projection).
• Linear Precision
• A high degree Bézier curve can be forced into a
  straight line segment by placing the control
  points on a straight line.
• Variation Diminishing
• The curve does not oscillate unduly.

33
Exercise Bézier Circles

• Question
• Create a single closed cubic Bézier curve
  that approximates a circle.

• Answer
• (a) Closed curve (b)
  continuous at (c) All
  control points are collinear which implies, by
  the convex hull property, that
  . (d) This is a point
  not a circle.

34
The de Casteljau Algorithm

• A recursive in-betweening (linear interpolation)
  process which can be used to
• Generate a point on the curve (the alternative is
  to evaluate the Bézier curve equation).
• Split a Bézier curve into two halves.
• Working from the initial control polygon (the
  line segments joining adjacent control points)
• a point is placed of the way along
  each edge.
• These points are joined into a new
  control polygon.
• The process is repeated until only a
  single point is introduced.
• This is the point at parameter
  on the curve.

35
de Casteljau Specifics

• The recurrence relation is
• Where are the original control
  points and is the point on the
  curve.
• The intermediate control points constitute two
  Bézier curves split from the original at

36
The Problem of Local Control

• Bézier curves are geometrically intuitive but
  they fail to provide local control.
• A designer cannot concentrate on altering part of
  a Bézier curve without affecting the rest.
• This is because the basis functions, which define
  the weight (influence) given to each control
  point, are all non-zero over . So, each
  control point has some influence over the shape
  of the whole curve.
• Solution build a curve by piecing together
  polynomial segments.
• This is the purpose behind B-spline curves.

37
Evaluation
38
Surfaces

• Curves generalise to Surfaces
• A surface is a function of two variables
  rather than just one
  .
• A rectangular surface has a curve running along
  each of its four edges.
• A rectangular surface can be envisioned as a
  curve which changes its shape as it is swept
  through space. Each control point of the initial
  curve is moved along a path which is itself a
  curve.
• Surface Definition
• An -degree by -degree Bézier patch is
  defined by control points
  
  

39
Surface Properties

• Adjacent control points can be connected to form
  a control net (which serves the same purpose as a
  control polygon).
• The degrees along and do not have to
  match.
• B-spline surfaces are a similar extension of
  B-spline curves.
• Continuity between patches
• C0 continuous in position. The four edge c.p.
  must match.
• C1 continuous in position and tangent vector.
  The two control points on either side of each
  edge c.p. must be colinear with the edge point
  and each other and be equidistant from the edge
  point.
• Algorithms (e.g. de Casteljau subdivision) and
  properties (e.g. convex hull) still apply with
  some small modification.

40
Example Surfaces

• Example cubic Bézier patch
• The four corner c.p. are interpolated.
• Each edge is a curve defined by four c.p.
• There are four interior c.p. determining the
  inner shape.

41
Exercise Designing Surfaces

• Question
• Specify the control points of an open-ended
  continuous cylinder with radius and
  height composed of two patches. You
  need only create one patch and explain how the
  other follows.

42
Rendering Bézier Surfaces

• Adaptive Subdivision Approximate the surface
  with planar polygons,
• Check if a quadrilateral with corners at the
  endpoints is an adequate approximation to the
  surface.
• If so, halt recursion.
• If not, subdivide into two smaller Bézier
  patches using bi-parametric de Casteljau.
  Subdivision must alternate in and on
  successive recursive subdivisions.
• Need to specify a tolerance for when the
  quadrilateral is an adequate approximation.
• The set of resulting (non-planar) quadrilaterals
  is further subdivided into (planar) triangles.

43
Surface Normals

• Often require a surface normal (e.g. Gouraud and
  Phong shading)
• Procedure
• Given a surface
• Calculate the partial derivatives and
  . These are tangents to the surface.
• Take the cross product to get
  the normal
• Scale the normal so that it is a unit vector

44
Subdivision Surfaces

• Motivation
• Creating objects from Bézier patches is very
  difficult if they have a non-planar topology. For
  instance a C1 sphere cannot be constructed from
  non-degenerate patches. But complex topologies
  and many-sided patches are possible with
  subdivision surfaces,
• Method
• Similar to Chaikins corner cutting.
• Start with a closed control net roughly
  approximating the object.
• Successively refine the control net by
  introducing new c.p.. In the limit this
  produces a smooth surface.
• Proviso
• There may be exception points where continuity is
  not as high.

45
Doo-Sabin Subdivision

• Mostly produces a quadratic B-spline surface.
• Algorithm
• Input an arbitrary open or closed polyhedron
  Output a smooth surface.
• Refinement
• Find the centroid of each face (average its
  vertices).
• Find the midpoints of all straight lines
  connecting the centroid to
  its defining vertices.
• Construct a new polyhedron from these midpoints
  
  by forming
• E-faces. For a given edge there are four
  midpoints controlled by its endpoints.
  Connect these midpoints into
  a rectangle.
  
• V-faces. Form a face with the midpoints that
  surround an
  original vertex.
• F-faces. Connect the midpoints belonging to the
  same face.

46
Curves and Surfaces in OpenGL

• use Evaluators to compute points on a curve or
  surface. A very powerful mechanism.
• setup glMaplf(type, u_min, u_max, stride, order,
  point_array)
• example glMaplf(GL_MAP_VERTEX_3, 0.0, 1.0, ?, 4,
  data)
• enable glEnable(type)
• rendering glEvalCoordlf(u) replaces glVertex
• Similar calls for creating surfaces.