Презентация на тему: Topic 3

Topic 3. Reminder: 2 D Curves Curves in 3 D Surfaces in 3 D Surface Example: Planes in 3 D Surface Example: Planes in 3 D Topic 3. Polygonal Meshes Polygonal Meshes Why use polygon meshes? Polygon Meshes Topic 3. Controlling a curve Interpolation with Splines Cubic Curve Cubic Hermite Interpolation Cubic Hermite interpolation Cubic Hermite Interpolation Matrix Representation Matrix Representation Matrix Representation of Polynomials Matrix Representation of Polynomials Hermite Basis Function Catmull -Rom Interpolation Topic 3. Topic 3. Topic 3. Topic 3. Topic 3.
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Первый слайд презентации: Topic 3

3D Objects General curves & surfaces in 3D Polygonal Meshes Interpolation algorithms

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Слайд 2: Reminder: 2 D Curves

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Слайд 3: Curves in 3 D

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Слайд 4: Surfaces in 3 D

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Слайд 5: Surface Example: Planes in 3 D

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Слайд 6: Surface Example: Planes in 3 D

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Слайд 7: Topic 3

3D Objects General curves & surfaces in 3D Polygonal Meshes Interpolation algorithms

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Слайд 8: Polygonal Meshes

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Слайд 9: Polygonal Meshes

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Слайд 10: Why use polygon meshes?

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Слайд 11: Polygon Meshes

Set of edge-connected planar polygons (usually triangles or quads) Faces share vertices and edges To avoid repeating vertices, store each vertex once Each face stored as set of indices into the vertex list Connectivity of faces also called mesh topology Normal at vertex often estimated as average of unit normals of all faces sharing that vertex Useful in practice, but less precise than differentiating original surface

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Слайд 12: Topic 3

3D Objects General curves & surfaces in 3D Polygonal Meshes Interpolation algorithms

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Слайд 13: Controlling a curve

Specify control parameters at a few locations Points Tangents … Make the curve conform to these parameters

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Слайд 14: Interpolation with Splines

Want: Smooth curve through sequence of points Intuition: Generate the curve in parts, one between each pair of points This is called a spline curve Has local control (small change won’t affect whole curve)

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Слайд 15: Cubic Curve

P(t ) = at 3 + bt 2 + ct + d 4 degrees of freedom For instance, can be specified completely by 4 points on the curve Popular tradeoff between control and simplicity Multiple cubic segments can be linked together into a longer and more complex curve

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Слайд 16: Cubic Hermite Interpolation

Specify positions h 0, h 1 and tangents (slopes, derivatives) h 2, h 3 at two points: t = 0 and t = 1

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Слайд 17: Cubic Hermite interpolation

Q: Why tangents and not two extra points? A: When we want two curve segments to link up smoothly, we can just require them to have a common tangent at the boundary

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Слайд 18: Cubic Hermite Interpolation

P(t ) = at 3 + bt 2 + ct + d P’(t ) = 3at 2 + 2bt + c h 0 = P(0) = d h 1 = P(1) = a + b + c + d h 2 = P’(0) = c h 3 = P’(1) = 3a + 2b + c

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Слайд 19: Matrix Representation

h 0 = P(0) = d h 1 = P(1) = a + b + c + d h 2 = P’(0) = c h 3 = P’(1) = 3a + 2b + c

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Слайд 20: Matrix Representation

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Слайд 21: Matrix Representation of Polynomials

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Слайд 22: Matrix Representation of Polynomials

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Слайд 23: Hermite Basis Function

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Слайд 24: Catmull -Rom Interpolation

Want: Smooth curve through sequence of points Intuition: A plausible tangent at each point can be inferred directly from the data Now use Hermite Interpolation

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