CSC418 - Notes
Topic 11: Curves

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Key concepts & Readings

General Readings (Hill: c. 11. Foley & van Dam: p. 478-516)
Continuous Curves
- Parametric, Explicit, Implicit and Representations
- Polynomials
- Continuity: C and G - Position, Tangentional... (Hill: p. 601)
2D Parametric Curves (Shirley: p. 36)
3D Parametric Curves (Shirley: p. 41-42)
Hermite Curves (Shirley: p. 238-240)
Basis Functions
Basis Matrix
Bezier Curves (Shirley: p. 232-236. Hill: 609-617)
- Bernstein's Polynomials
Splines (Shirley: p. 240-245. Hill: p. 618-633. Hill: 605-606)
Curve Properties
- Affine Invariance
- Variation Diminishing
- Convex Hull
- Local Control

OpenGL

Notes

2D - Curves

Curves have implicit, explicit, and parametric representations. Parametric represenations are the most common in computer graphics.

Implicit Representation

An implicit equation in 2D:

All points that satisfy the above equation lie on the curve.

Points can not be immediately calculated
does have values for all
divides the space into positive and negative regions, e.g., all points for which are said to belong to the positive region of the space. Similarly points for which are said to belong to the negative region of the space.
The gradient vector evaluated at a point $p=(x_{p},y_{p})$ of the implicit curve is perpendicular to the tangent vector of the curve at that point. It is also called the normal vector to the curve at that point.

Explicit Representation

i.e., given x (abscissa) find y (ordinate)

Parametric Representation

MATH or in vector notation . The curve said to be parametrized by .

Easy to generate points along the curve by evaluating and at a sequence of values, so such curves can be drawn quickly.

Tangent for the parametric form

How to find tangent of parametric curve at point $(x_{0},y_{0})$ :

First, find the value for that gives rise to the point $(x_{0},y_{0})$ . Then evaluate at that value for .

Reparameterization

Parameterizations are in general not unique. Consider the following parameterizations for a line:

  L(P₀,P₁) = P₀ + u(P₁-P₀),  u = [0...1]
  L(P₀,P₁) = v(P₁-P₀)/2 + (P₁+P₀)/2,  v = [-1...1]

Parameterizations can be changed to lie between desired bounds. To reparameterize from u = [a...b] to w = [0...1], we can use w = (u-a)/(b-a), which gives u = w(b-a) + a. Thus, we have:

  P(u), u = [a...b]   =    P(w(b-a)+a), w = [0...1

2D Lines

A line can be defined between two points $(x_{0},y_{0})$ and $(x_{1},y_{1})$ , by a single point and a direction vector $(d_{x},d_{y})$ , or by a point and a normal vector $(n_{x},n_{y})$ . The three cases are infact identical because:

or $(d_{y},-d_{x})$ (Exercise: rotate vector by 90 degrees using the 2D rotation matrix).

Explicit Representation of a 2D Line

, where is the slope and is the y-intercept. This equation can not represent any line parallel to y-axis as such lines have inifinte slope.

Implicit Representation of a 2D Line

Form 1:

Converting slope-intercept for into implicit form:

Form 2:

Equation of line between two points $(x_{0},y_{0})$ and $(x_{1},y_{1})$ is .

Derivation of an equation of a line given two points $(x_{0},y_{0})$ and $(x_{1},y_{1})$

Eq. of a line has the form .

Gradient vector of the line:

Now, a vector along the line is:

We remember that gradient of an implicit curve is the normal vector, so

Therefore,

And

Point $(x_{0},y_{0})$ (and $(x_{1},y_{1})$ ) lies on the line so it must satisfy the equation of the line, so

(Note. in book this is incorrect)

Distance between a point and a line:

A line divides the space into two halves: +ve & -ve. Distances on one side of the line are positive, those on the otherside of the line are negative. The choice of the positive side and negative side is arbitrary because is equivalent to . For most problems, usually there are advantages of choosing a particular side as being positive.

Signed distance from a line to a point is:

Parametric form of a 2D Line

Given two points and . Let .

Line: MATH or in vector notation , where $t\in \lbrack 0,t]$ .

Ray: , where $t\geq 0$ .

Line segment between $(x_{0},y_{0})$ and $(x_{1},y_{1})$ : , where $0\leq t\leq 1$ .

Intersection of two lines:

Given two lines: and .

For two lines to intersect the following must hold for some values of and :

The above is a system of two equations with two unknowns. We can solve for (or ) and then substitute the value in the first line (or second) to get the intersection point. The system of equation has no solution if the lines are parallel to each other.

Details:

The point of intersection is:

Intersection Line, Line-segments and rays:

The above approach can also be used for finding intersection between:

line(t) - ray(u) intersection: $t\in R$ and $u\geq 0$ .
line segment(t) - ray(u) intersection: $0\leq t\leq 1$ and $u\geq 0$ .
line segment(t) - line segment(u) intersection: $0\leq t\leq 1$ and $0\leq u\leq 1$ .
line(t) - line segment(u) intersection: $t\in R$ and $0\leq u\leq 1$ .
ray(t) - ray(u) intersection: $t\geq 0$ and $u\geq 0$ .

Solve for and and check the bounds.

The Circle

Circle of radius and center $(x_{c},y_{c})$ .

Implicit form

Parametric form

. Here is the angle from the x-axis.

Ellipses

Implicit form

Centered at origin and axis aligned:

General form (conics): , where $ac-b^{2}>0$ .

Parametric form

Note: unlike circles, parameter , is not the angle with the x-axis.

3D - Lines (Parametric Representation)

Given two points

and . Let .

Line: MATH or in vector notation , where $t\in \lbrack 0,t]$ .

Ray: , where $t\geq 0$ . (See also, Ray Tracing).

Line segment between $(x_{0},y_{0})$ and $(x_{1},y_{1})$ : , where $0\leq t\leq 1$ .

We can use similar reasoning as shown for 2D parametric lines to find intersections for 3D parametric lines.

3D - Surfaces

tangent plane to a surface at $\QTR{bf}{p}$ is a plane containing tangent vectors, at $\QTR{bf}{p}$ , to all curves on the surface that go through $\QTR{bf}{p}$ .
surface normal is the vector perpendicular to the tangent plane

Implicit form

surface normals: $\nabla f(x,y,z)$

Parametric form

let then
tangent vector at $(u_{0},v_{0})$ are: and .
together these tangent vectors define the tangent plane at $(u_{0},v_{0})$ .
the surface normal is .

Parametric Sphere

Planes

Implicit Form:

Plane through $\QTR{bf}{p}_{0}$ and perpendicular to $\QTR{bf}{n}$ .

Plane can be uniquely defined be three non-collinear points , , and .

$n=a\times b$

Parametric form

Planar Patch

Rhombus defined by bounds on and .

Bilinear Patches

Defined by 4 points, no 3 of which are colinear

, where . (Shirley p. 231)

Derivation:

given $\QTR{bf}{p}_{00}$ , $\QTR{bf}{p}_{01}$ , $\QTR{bf}{p}_{10}$ , $\QTR{bf}{p}_{11}$

define and

then connect with a line

Substitute $l_{0}$ and $l_{1}$ to get the above form.

Surfaces of Revolution

revolve a curve in x-z plane, , about z-axis
so each point on $\QTR{bf}{c}(v)$ traces out a cirle parallel to x-z plane with radius
let is the parameter of revolution then the rotated profile is (hint: circle's parametric form is )

An example

Rotating a circle with radius and center $(c_{x},c_{z})$ about z-axis gives rise to a Torus (like a donut).

Quadratic Bezier Curve

for $t\in \lbrack 0,1]$

$w_{0}=(1-t)^{2}$

$w_{1}=2(1-t)t$

$w_{2}=t^{2}$

$w_{i}$ are positive and MATH .

Test:

Quadratic Bezier Surface

to be written

Parametric Cubic Curves

Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.

A parametric cubic curve in 3D is defined by:

Usually, we consider t = [0...1].

A compact version of the parametric equations can be written as follows:

Similarly, we can write

  y(t) = T B
  z(t) = T C

Each dimension is treated independently, so we can deal with curves in any number of dimensions.

The derivatives of the curve with respect to t can be expressed as follows:

  x'(t) = [3t²  2t  1  0] A

It is often convenient to think of the parameter t as being time in order to visualize some of the properties of the curve. The derivatives also have an intuitive interpretation in the cartesian space of the curve:

A First Example

Suppose we wish to define a cubic curve such that the user specifies the position of two endpoints and a midpoint, as well as the tangent at the midpoint. The following figure illustrates some sample curves.

We'll first construct the cubic curve for x(t). There are four constraints that are to be met and four unknowns:

We can solve for A using A = B^-1x. The final equation for x is thus:

  x(t) = T B^-1 G_x

The matrix B^-1 is often called the basis matrix, which we shall denote by M. We can thus write

  x(t) = T M G_x

In this case, we have

      [ -4  0 -4  4 ]
  M = [  8 -4  6 -4 ]
      [ -5  4 -2  1 ]
      [  1  0  0  0 ]

Lastly, we can also write

  x(t) = [ f₁(t) f₂(t) f₃(t) f₄(t) ] G_x

where f₁(t) ... f₄(t) are the functions obtained from the product T * M. These are functions are called the blending or basis functions, because they serve to give a weighting to the various components of the geometry vector, G. In this case, these are

f₁(t) = -4t³ + 8t² - 5t + 1
f₂(t) = -4t² + 4t
f₃(t) = -4t³ + 6t² - 2t
f₄(t) =  4t³ - 4t² + 1,

where f₁(t) is the weighting function for P₀, f₂(t) is the weighting function for P_0.5, f₃(t) is the weighting function for T_0.5, and f₄(t) is the weighting function for P₁. These basis functions look as follows:

The curves for y(t) and z(t) are contructed in an analogous fashion to that for x(t). The basis matrix and basis functions thus remain the same. Only the geometry vector changes. The geometry vector for y(t) gives the y components of P₀, P_0.5, T_0.5, and P₁. In general, we can write the curve as a single vector equation

P(t) = T M G

which encompasses the following three equations:

x(t) = T M G_x
y(t) = T M G_y
z(t) = T M G_z

Hermite Curves

As a second example, let's look at Hermite curves. Hermite curves are defined by two points and two tangent vectors.

Let's derive the equation for Hermite curves using the following geometry vector:

G_h = [ P₁ P₄ R₁ R₄ ]^T

As before, we'll express the curve as:

x(t) = T A_h
     = T M_h G_h

The constraints we'll use to define the curve are:

x(0)  = P₁ = [ 0 0 0 1 ] A_h
x(1)  = P₄ = [ 1 1 1 1 ] A_h
x'(0) = R₁ = [ 0 0 1 0 ] A_h
x'(1) = R₄ = [ 3 2 1 0 ] A_h

Writing these constraints in matrix form gives:

G_h = B_h A_h
A_h = (B_h)^-1 G_h
x(t) = T A_h
     = T (B_h)^-1 G_h
     = T M_h  G_h

The inverse of B_h is thus defined as the basis matrix for the hermite curve.

      [  2 -2  1  1 ]
M_h  = [ -3  3 -2 -1 ]
      [  0  0  1  0 ]
      [  1  0  0  0 ]

As before, the basis functions are the weighting factors for the terms in the geometry vector, and are given by the product T M_h. Thus, for basis functions for Hermite curves are

f₁(t) =  2t³ - 3t² + 1
f₂(t) = -2t³ + 3t²
f₃(t) =   t³ - 2t² + t
f₄(t) =   t³ -  t²

These basis functions look as follows:

Bezier Curves

Bezier curves are a variation of the Hermite curves. They are specified by four points:

The curve is closely related to the Hermite curve:

P_1h = P₁
P_4h = P₄
R_1h = 3 (P₂ - P₁)
R_4h = 3 (P₄ - P₃)

We'll arrive at the basis matrix by expressing the above in matrix form:

[ P_1h ]   [  1 0  0 0 ] [ P₁ ]
[ P_4h ] = [  0 0  0 1 ] [ P₂ ]
[ R_1h ]   [ -3 3  0 0 ] [ P₃ ]
[ R_4h ]   [  0 0 -3 3 ] [ P₄ ]

P_h = M_bh P_b

The equation for a bezier curve can thus be written as:

P(t) = T M_h M_bh P_b
P(t) = T M_b P_b

where

      [ -1  3 -3  1 ]
M_b  = [  3 -6  3  0 ]
      [ -3  3  0  0 ]
      [  1  0  0  0 ]

The Bezier basis functions are as follows:

P(t) = T M_b G_b
     = f₁(t) P₁ + f₂(t) P₂ + f₃(t) P₃ + f₄(t) P₄
where
  f₁(t) =  -t³ + 3t² - 3t + 1
  f₂(t) =  3t³ - 6t² + 3t
  f₃(t) = -3t³ + 3t²
  f₄(t) = t³

or alternatively, 

  f₁(t) = (1-t)³
  f₂(t) = 3t (1-t)²
  f₃(t) = 3t² (1-t)
  f₄(t) = t³

Convex hull property

The convex hull property ensures that the curve will never pass outside of the convex hull formed by the four control vertices. As such, it lends a measure of predictability to the curve. The Bezier basis functions satisfy the conditions necessary for the convex hull property, namely:

0 <= f_i(t) <= 1 for t in [0,1].
f₁(t) + ... + f_n(t) = 1

Bezier Curves of degree n

It is not per chance that the basis functions for Bezier curves have the convex hull property. How might one then go about designing a set of basis functions that sum to one and ensure that each individual basis function remain in the range [0,1]? The Bernstein polynomials have exactly the required properties.

Using the Bernsteinn polynomials, we can construct a Bezier curve of arbitrary degree. For curves of higher degree than the cubic Bezier curve discussed thus far, we'll need more than four control points. The general Bezier curve of degree n is given by

The basis functions are equivalent to the terms arising from the expansion of

using the binomial expansion theorem. It is also obvious from this why the sum of all the terms is equal to one.

Piecewise Hermite and Bezier Curves

Hermite curve segments can be connected in a continuous fashion by ensuring that the end-point of one curve is the same as the starting point of another, as well as ensuring that the tangent vectors for this point have the same direction.

In terms of the above figure, this means P₁'=P₄ and R₁'=k R₄ . For a Bezier curve, the conditions are that the the last two points of one curve and the first two points of the second curve are aligned.

Geometric and Parametric Continuity

Geometric Continuity

G₀: curves are joined
G₁: first derivatives are proportional at the join point
The curve tangents thus have the same direction, but not necessarily the same magnitude.
i.e., C₁'(1) = (a,b,c) and C₂'(0) = (k*a, k*b, k*c).
G₂: first and second derivatives are proportional at join point

Parametric Continuity

C₀: curves are joined
C₁: first derivatives equal
C₂: first and second derivatives are equal
If t is taken to be time, this implies that the acceleration is continuous.
C_n: nth derivatives are equal

As their names imply, geometric continuity requires the geometry to be continuous, while parametric continuity requires that the underlying parameterization be continuous as well.

Parametric continuity of order n implies geometric continuity of order n, but not vice-versa.

Summary of Bezier and Hermite Curves

offer local control
offer C₁ continuity
interpolates (some) control points

Splines

splines are cubic curves which maintain C₂ continuity.
natural spline
- interpolates all of its control points.
- equivalent to a thin strip of metal forced to pass through control points
- no local control
B-spline
- local control
- does not interpolate control points

The following is an example of a five-segment B-spline curve (although this is simply a hand-drawn example). The points which indicate the ends of the individual curve segments and thus the join points are known as the knots.

Each curve segment is determined by four control points, as follows:

B-spline curves are defined by a basis matrix, just like the other types of cubic curves. We shall not discuss the derivation of this matrix here.

                [ -1  3 -3 1 ]
M_bspline    = 1/6 [  3 -6  3 0 ]
                [ -3  0  3 0 ]
                [  1  4  1 0 ]

where the geometry vector consists of four consecutive control points. It is easy to show that B-spline curves are C2 continuous and that they satisfy the convex-hull property. The basis functions for B-splines are as follows:

The general form of a multi-segment B-spline can be written as follows:

Multiple knots

Control vertices can be repeated in order to allow for reduced geometric continuity at the join points.

multiplicity
1	G₂ continuous
2	G₁ continuous
3	G₀ continuous, interpolates point

NURBS: Nonuniform Rational B-splines

One of the disadvantages of the curves discussed to date is that they cannot be used to create common conic shapes such as circles, ellipses, parabolas, etc. This can be done using rational cubic curves, however. A rational cubic curve segment in 3D can be constructed as follows

  x(t) = X(t)/W(t)
  y(t) = Y(t)/W(t)
  z(t) = Z(t)/W(t)

where each of X(t), Y(t), Z(t), and W(t) are cubic polynomial curves. Defining curves as rational polynomials in this manner allows for simple exact represenations of conic sections such as circles, as well as curves which are invariant under perspective projection.

Curve Properties

Affine Invariance

to be written

Variation Diminishing

to be written

Convex Hull

Explained earlier

Local Control

Explained earlier

Exercises

(to be written)

CSC418 - Notes Topic 11: Curves

Key concepts & Readings

OpenGL

Notes

2D - Curves

Implicit Representation

An implicit equation in 2D:

Explicit Representation

Parametric Representation

Tangent for the parametric form

How to find tangent of parametric curve at point :

Reparameterization

2D Lines

Explicit Representation of a 2D Line

Implicit Representation of a 2D Line

Form 1:

Form 2:

Derivation of an equation of a line given two points and

Distance between a point and a line:

Parametric form of a 2D Line

Intersection of two lines:

Details:

Intersection Line, Line-segments and rays:

The Circle

Implicit form

Parametric form

Ellipses

Implicit form

Parametric form

3D - Lines (Parametric Representation)

3D - Surfaces

Implicit form

Parametric form

Parametric Sphere

Planes

Implicit Form:

Parametric form

Planar Patch

Bilinear Patches

Derivation:

Surfaces of Revolution

An example

Quadratic Bezier Curve

Quadratic Bezier Surface

to be written

Parametric Cubic Curves

A First Example

Hermite Curves

Bezier Curves

Convex hull property

Bezier Curves of degree n

Piecewise Hermite and Bezier Curves

Geometric and Parametric Continuity

Summary of Bezier and Hermite Curves

Splines

Multiple knots

NURBS: Nonuniform Rational B-splines

Curve Properties

Exercises

CSC418 - Notes
Topic 11: Curves

How to find tangent of parametric curve at point $(x_{0},y_{0})$ :

Derivation of an equation of a line given two points $(x_{0},y_{0})$ and $(x_{1},y_{1})$