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Curved Surfaces

Sweep Representations

Constructive Geometry

Bezier Curves

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Sweep representations

Sweep representations are used to construct 3D objects that have some kind

of symmetry.

For example, a prism can be generated using a translational sweep

Rotational sweeps can be used to create curved surfaces like an ellipsoid or

a torus

In both cases, you start with a cross-section and generate more vertices by

applying the appropriate transformation.

More complex objects can be formed by using more complex transformations.

Also, you can define a path for a sweep that allows you to create more inter-

esting shapes.

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Translational Sweep

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Rotational Sweep

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Constructive Solid Geometry

Another approach to object modeling consists of using set operations to com-

bine two objects.

Union

Intersection

Difference

Generally start with a set of primitives that serve as building blocks.

Ray casting is a method used for determining boundaries of the resulting

object if you start with a boundary representation.

Octree representations are designed to make this process easier.

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Constructive Geometry Example

Figure b is the intersection of the original solids

Figure c is the difference between the two original solids

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Ray Casting

Operation Surface Limits

Union A, D

Intersection C, B

Difference (obj2 obj1) B, D

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Quadtree and Octree Representations

The basic idea is to divide space up into homogeneous regions.

Use a tree structure with large regions near the root

If a region is uniform, store its properties

If a region is non-uniform, subdivide it and repeat the process

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Curves

Curves are used in graphics for a number of purposes

to represent surfaces (usually tessellated)

to represent paths for motion

Splines approximate curves with a series of polynomial segments

There are a number of common approaches

Bezier curves

Cubic splines

B-Splines

Rational splines

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Geometric Algorithm for Bezier curves

P0

A B

C

B1C1A1

P3

P2P1

Start with endpoints and two control points

find midpoints of lines joining these points

find midpoint of lines joining points generated in 2

find midpoint of line joining points generated in 3

Recursively apply algorithm to two new sets of points until segments are short

enough to make a smooth curve

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Parametric Curves

A curve can be represented as a set of functions that represent x and y values

as a function of some parameter, t the parametric representation of a circle is

x(t) = r cos t

y(t) = r sin t

each value of t generates one point on the curve

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Parametric representation of Bezier curves

For a third-order Bezier curve, we can represent the curve as

B(t) =

x

y

= (1 t)3P0+t(1 t)2P0+t2(1 t)P2+t3P3

B(t) = P0 and B(t) = P3 so we have a cubic curve which goes through

P0 and P3. The vectors P0P1 and P2P3 are tangent to the curve at the

endpoints.

P0

A B

C

B1C1A1

P3

P2P1

This property of the tangents guarantees that the recursively generated curve

is smooth

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Curved Surfaces

The same algorithms can be used for curves in three dimensions just add

z-coordinates to the equations