Limitations of Polygonal Meshes

Planar facets

Fixed resolution

Deformation is difficult

Representing Polygon Meshes

1). The explicit way: just list 3D vertices of each polygon in a certain

order. The problems are, firstly it represents same vertex many

times and secondly, no explicit representation of shared edges

and vertices

2). Pointer to a vertex list: store all vertices once into a numbered

list, and represent each polygon by its vertices. It saves space

(vertex only listed once) but still has no explicit representation of

shared edges and vertices

3). Explicit edges: list all edges that belong to a polygon, and for

each edge list the vertices that define it along with the polygons

of which it is a member.

)),,(),...,,,(),,,(( 222111 nnn zyxzyxzyxP

)5,4,3,1(P

),,( 121 PVVE

Types of Curves

1). Explicit:

In the Cartesian plane, an explicit equation of a planar curve is

given by y = f(x). The difficulties with this approach are that

(1) it is impossible to get multiple values of y for a single x, so

curves such as circles and ellipses must be represented by

multiple curve segments; and (2) describing curves with vertical

tangents is difficult and numerically unstable.

2). Implicit:

f(x, y) = 0 Ax+By+C =0

This method has difficulties on determining tangent continuity of

two given curves which is crucial in many applications.

(Circle can be defined as: x2+y2=1, but what about a half circle?)

Types of Curves

3). Parametric Curves:

The cubic polynomials that defines a curve segment Q(t)= [x(t)

y(t)]T are of the form

Written in matrix form, it becomes

Q(t)= [x(t) y(t)] = T C

where

yyyy

xxxx

dtctbtaty

dtctbtatx

23

23

)(

)(

x

y

y

y

y

x

x

x

d

c

b

a

d

c

b

a

C 123 tttT

Continuity

The derivative of Q(t) is the parametric tangent vector of the curve.

Applying the definition to the curves equation, we have

Often we will want to represent a curve as a series of curves pieced

together. But if we will want these curves to fit together reasonably ...

continuity!

yyyxxx ctbtactbta

CttCTdt

d

dt

dy

dt

dxtQtQ

dt

d

2323

]0123[

)(')(

22

2

Continuity

Two curve segments

join together: G0

geometric continuity.

The directions of the tangent

vectors (not necessarily the

magnitudes) are equal: G1

geometric continuity.

Both the directions and

magnitudes are equal: C1

parametric continuity.

Second-order parametric

continuity: C2 parametric

continuity.

Note: For two curves to join smoothly, we require only that their tangent-vector

directions match; we do not require that their magnitudes match.

Specifying Curves

Interpolating spline

curve passes through the knots

knots

control points that lie on the

curve

Control points

a set of points that influence

the curve's shape

Approximating spline

control points merely influence

shape (Hermite Curves, B-spline)

Specifying curves

Control Points: A set of points that influence

the curves shape.

Knots: Control points that lie on the

curve.

Interpolating spline: Curve passes through the

control points.

Approximating spline: Control points merely influence

shape.

Hermite Spline

A spline is a parametric curve defined by control points

The term spline dates from engineering drawing, where a spline was a piece of flexible wood used to draw smooth curves

The control points are adjusted by the user to control the shape of the curve

A Hermite spline is a curve for which the user provides:

The endpoints of the curve

The parametric derivatives of the curve at the endpoints (tangents with length)

The parametric derivatives are dx/dt, dy/dt, dz/dt

That is enough to define a cubic Hermite spline, more derivatives are required for higher order curves

Hermite Spline (2)

Say the user provides

A cubic spline has degree 3, and is of the form:

For some constants a, b, c and d derived from the control points, but how?

We have constraints:

The curve must pass through x0 when t=0

The derivative must be x0 when t=0

The curve must pass through x1 when t=1

The derivative must be x1 when t=1

dctbtatx 23

1010 ,,, xxxx

Bzier Curves

Hermite cubic curves are difficult to model need to specify point and gradient.

More intuitive to only specify points.

Pierre Bzier specified 2 endpoints and 2 additional control points to specify the gradient at the endpoints.

Can be derived from Hermite matrix: Two end control points specify tangent

Bezier Curve Properties

The first and last control points are interpolated

The tangent to the curve at the first control point is along the line joining the first and second control points

The tangent at the last control point is along the line joining the second last and last control points

The curve lies entirely within the convex hull of its control points

The Bernstein polynomials (the basis functions) sum to 1 and are everywhere positive

They can be rendered in many ways

E.g.: Convert to line segments with a subdivision algorithm

Rendering Bezier Curves (1)

Evaluate the curve at a fixed set of parameter values and join the points with straight lines

Advantage: Very simple

Disadvantages:

Expensive to evaluate the curve at many points

No easy way of knowing how fine to sample points, and maybe sampling rate must be

different along curve

No easy way to adapt. In particular, it is hard to measure the deviation of a line segment from

the exact curve

Convex Hull Property

If we take all of the control points for a Bezier curve and construct a convex polygon around them, we have the

convex hull of the curve

An important property of Bezier curves is that every point on the curve itself will be somewhere within the convex

hull of the control points

p0

p1

p2

p3

Connecting Bezier Curves

A simple way to make larger curves is to connect up Bezier curves

Consider two Bezier curves defined by p0p3 and v0v3 If p3=v0, then they will have C

0 continuity

If (p3-p2)=(v1-v0), then they will have C1 continuity

C2 continuity is more difficult

p0

p0

p1

p2

P3

P3

p2

p1 v0

v1

v2

v3

v3

v2

v1

v0

C0 continuity C1 continuity

A Bzier Curve

For example, if we are given four control points (0,0), (0.2,0.8), (0.9,

0.1), (1,1). Then we have

And the parametric equation will be

At any point on this curve, the tangent vector to the curve is

00

4.26.0

5.45.1

1.31.1

11

1.09.0

8.02.0

00

0001

0033

0363

1331

yx

yx

yx

yx

dd

cc

bb

aa

tttty

ttttx

4.25.41.3)(

6.05.11.1)(

23

23

00|]

)(,

)([|)(' tttt

dt

tdy

dt

tdxtQ

There are nonuniform, nonrational B-splines, which differ from the uniform, nonrational B-Splines in that the Uniform B-splines have knots at equal intervals in t. The distances in t between adjacent knots are the same, and the blending functions for each curve segment are the same.

In nonuniform B-Splines, the parametric intervals between knots are not equal, and therefore the blending functions are no longer the same for each interval, but vary from curve segment to curve segment.

These curves have several advantages over uniform B-splines:

Continuity of join points can be reduced from C2 to C1 to C0 to none. If the continuity is reduced to none, then the curve interpolates a control point, but without the effect of uniform B-splines, where the curve segments on either side of the control point are straight lines.

The starting and ending points can be interpolated exactly, without introducing straight line segments.

It is possible to add an additional knot and control point so that the resulting curve can be easily reshaped, whereas this cannot be done with uniform B-splines.

A Bezier curve lies within a convex hull of the control polygon Convex Hull the minimum convex polygon enclosing a set of given points

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

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

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.