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Constructive Solid Geometry Mel Slater

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Introduction Quadric Surfaces Ray Intersections with Quadrics Set Operations Ray Classification

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Quadric Surfaces Consider a sphere And also a cylinder: These are special cases of a general class of object called quadrics.

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Quadric Surfaces Let Q be a 4*4 matrix Let p = (x,y,z,1) s(x,y,z) = p T Qp = 0. The quadric is the boundary and the interior of the surface defined by the equation, and hence consists of p T Qp 0.

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Quadric Surfaces Note that the highest power of any coordinate is 2 (hence quadric). Note that the a,…,j are given constants Special cases a,…,j of define different types of surface. Multiplying out we get:

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Plane Sphere Cylinder Special Cases

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Cone Paraboloid

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More Properties Given a quadric the normal at point p is given by 2p T Q. If all the points p on a quadric are transformed, eg, q = pM then the result is also a quadric –where M is a transformation matrix.

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Ray Intersection The beauty of the quadric representation is that the unified representation requires only one ray-intersection algorithm. Let p(t) = p + tv, t 0 –Be the parametric equation of the ray that starts at p in direction p. Substitute p(t) into p T Qp = 0 and solve the quadratic equation for t. This will give –Complex results if no intersection –Two equal values of t if tangent –Two real values – the entry and exit points otherwise. –(What happens in the case of a plane?)

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Set Operations The quadrics represent solid objects –Unlike boundary representations for mesh and B-Spline approaches We can use set operations such as union, intersection and difference to combine such solids together.

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Regularising Operation Given a point p we define the ‘open ball’. A boundary point in any point-set S is a point p such that for any >0 B(p, ) will contain points in S and also in not-S. S is an open set if it does not contain its boundary. Closure(S) = S boundary(S)

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Regularising Operation Interior(S) = S – boundary(S) Regularisation(S) = closure(interior(S)) For any set operation it is the regularisation of the result that is required. A op* B = regularisation(A op B) –where op is any operator such as union, difference, intersection

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Example Intersection results in illegal set (a)Is the set, which is closed in (b) with interior in (c) and regularisation in (d)

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CSG Operations We can define set operations such as , , (not) and difference -. We can make formulae expressing more complex objects as combinations of simpler objects, eg –A B – C –((A B) C)-D Each can be expressed as a binary operator tree.

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((A B) C)-D A B C D - Each expression can be represented as a binary operator tree.

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CSG Data Structure typedef struct _csgtree –Operator op; –Quadric primitive; –struct _csgtree *right; –struct _csgtree *left; } CSGTreePtr; Each leaf node represents a primitive (usually a quadric primitive). ‘op’ can also be ‘leaf’ rather than a combination operator.

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Intersecting a Ray Pseudo code for ray-tree intersection

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Intersecting a Ray The result will be a sequence of intersection points along the ray, represented by the parametric values –t 0, t 1, t 2, …,t n –where there are n intersections Each individual intersection with a solid will result (typically) in two values.

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Combine Function Shows how the result of the left and right solids combine together. Note that the set operations therefore occur at a 2D level.

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Examples www-2.cs.cmu.edu/afs/cs/misc/rayshade/ all_mach/omega/doc/Examples/jpg/csg.jpg ccvweb.csres.utexas.edu/ccv/projects/VisualEyes/ visualization/domainpara/algebraic2/parameterization/ gallery/Reconstruction/Csg/

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Summary Unified representation for quadric surfaces provides a simplicity and elegance for a wide class of shapes. CSG provides a simple methodology for combinations via set operations of solids CSG does not require quadrics, it can work other types of solid representation.

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