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# Surfaces

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### Surfaces

1. 1. S U R F A C E S <ul><ul><ul><li>By Abdul Ghaffar </li></ul></ul></ul>
2. 2. Chapter 6: Polygonal Meshes <ul><li>6.4.5 Discretely Swept Surfaces of Revolution </li></ul><ul><ul><li>Place all spline points at origin, and use rotation for affine transformation. </li></ul></ul><ul><ul><li>Base polygon called profile </li></ul></ul><ul><ul><li>Operation equivalent to circularly sweeping shape about axis </li></ul></ul><ul><ul><li>Resulting shape called surface of revolution. </li></ul></ul>
3. 3. Chapter 6: Polygonal Meshes <ul><li>6.5.1 Representation of Surfaces </li></ul><ul><ul><li>Similar to planar patch P(u,v) = C + a u + b v </li></ul></ul><ul><ul><li>Generalize: P(u,v) = (X(u,v), Y(u,v), Z(u,v)) (point form). </li></ul></ul><ul><ul><li>If v constant, u varies: v-contour </li></ul></ul><ul><ul><li>If u constant, v varies: u-contour </li></ul></ul><ul><li>Implicit Form of Surface </li></ul><ul><ul><li>F(x,y,z)=0 iff (x,y,z) is on surface. </li></ul></ul><ul><ul><li>F(x,y,z)<0 iff (x,y,z) is inside surface </li></ul></ul><ul><ul><li>F(x,y,z)>0 iff (x,y,z) is outside surface </li></ul></ul>
4. 4. Polygonal Meshes <ul><li>6.5.6 Rules Surfaces </li></ul><ul><ul><li>Surface is ruled if, through every one of tis points, there passes at least one line that lies entirely on the surface. </li></ul></ul><ul><ul><li>Rules surfaces are swept out by moving a straight line along a particular trajectory. </li></ul></ul><ul><ul><li>Parametric form: P(u,v) = (1-v)P 0 (u) +vP 1 (u). </li></ul></ul><ul><ul><li>P 0 (u) and P 1 (u) define curves in 3D space, defined by components P 0 (u)=(X 0 (u),Y 0 (u),Z 0 (u)). </li></ul></ul><ul><ul><li>P 0 (u) and P 1 (u) defined on same interval in u. </li></ul></ul><ul><ul><li>Ruled surface consists of one straight line joining each pair of points P 0 (u’) and P 1 (u’). </li></ul></ul>
5. 5. Chapter 6: Polygonal Meshes <ul><li>Cones </li></ul><ul><ul><li>Ruled surface for which P 0 (u) is a single point </li></ul></ul><ul><ul><li>P(u,v) = (1-v)P 0 +vP 1 (u). </li></ul></ul>
6. 6. Chapter 6: Polygonal Meshes <ul><li>Cylinders </li></ul><ul><ul><li>Ruled surface for which P 1 (u) is a translated version of P 0 (u): P 1 (u) = P 0 (u) + d </li></ul></ul><ul><ul><li>=>P(u,v)= P 0 (u) + d v </li></ul></ul>
7. 7. Chapter 6: Polygonal Meshes <ul><li>6.5.8 The Quadric Surfaces </li></ul><ul><ul><li>3D analogs of conic sections </li></ul></ul>
8. 8. Chapter 6: Polygonal Meshes <ul><li>6.5.8 The Quadric Surfaces </li></ul>
9. 9. Chapter 6: Polygonal Meshes <ul><li>Properties of Quadric Surfaces </li></ul><ul><ul><li>Trace is curve formed when surface is cut by plane </li></ul></ul><ul><ul><ul><li>All traces of quadric surfaces are conic sections. </li></ul></ul></ul><ul><ul><li>Principal traces are curves generated when cutting planes aligned with axes. </li></ul></ul>