CMSC427 Parametric surfaces and polygonal meshes Note These - - PowerPoint PPT Presentation

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CMSC427 Parametric surfaces and polygonal meshes Note These - - PowerPoint PPT Presentation

Nov 01, 2023 1.13k likes •1.36k views

CMSC427 Parametric surfaces and polygonal meshes Note These slides are incomplete See accompanying PDF with detailed outline Will develop many equations in class Reading later to supplement Moving to 3D Polygonal meshes

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SLIDE 1

CMSC427 Parametric surfaces and polygonal meshes

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SLIDE 2
  • These slides are incomplete
  • See accompanying PDF with detailed outline
  • Will develop many equations in class
  • Reading later to supplement

Note

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SLIDE 3
  • Polygonal meshes
  • Set of standard shapes

in Blender

  • And how to create them
  • And store them
  • And draw them

Moving to 3D

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SLIDE 4
  • Blending of four 3D points
  • Ruled surface
  • Swept out by sequence of lines

Bilinear patch

P0 P1 P2 P3

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SLIDE 5
  • Blend simultaneously along two lines
  • P01 = t(P1-P0) + P0
  • P23 = t(P2-P3) + P2
  • Same t in [0,1]

Bilinear patch

P0 P1 P2 P3 P01 P23

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SLIDE 6
  • Blend simultaneously along two lines
  • P01 = tP1 + (1-t)P0
  • P23 = tP3 + (1-t)P2
  • Same t in [0,1]
  • Then blend between

the two lines

  • P = sP23 + (1-s)P01
  • P = s(tP1 + (1-t)P0) + (1-s)(tP3 + (1-t)P2)

Bilinear patch

P0 P1 P2 P3 P01 P23 P

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SLIDE 7
  • Questions
  • What order polynomial?
  • Convex combination?
  • What is drawn if t is constant?
  • What is drawn if s is constant?
  • P = s(tP1 + (1-t)P0) + (1-s)(tP3 + (1-t)P2)

Bilinear patch

P0 P1 P2 P3 P01 P23 P

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SLIDE 8
  • Questions
  • What order polynomial?
  • Convex combination?
  • What is drawn if t is constant?
  • What is drawn if s is constant?
  • P = s(tP1 + (1-t)P0) + (1-s)(tP3 + (1-t)P2)
  • P = stP1 + s(1-t)P0 + (1-s)tP3 + (1-s)(1-t)P2

Bilinear patch

P0 P1 P2 P3 P01 P23 P

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SLIDE 9
  • What’s happening in this

surface? Coons patch

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SLIDE 10
  • What’s happening in this

surface?

  • Blending two arcs
  • Is this a ruled surface?

Coons patch

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SLIDE 11
  • Blend four arbitrary curves
  • Here C1, C2, D1, D2

Coons patch

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SLIDE 12

Circle with trig: review

Parametric equation

t

𝑦 = 𝑆 cos (𝑢) 𝑧 = 𝑆 sin (𝑢)

cos (𝑢) sin (𝑢)

0 ≤ 𝑢 ≤ ? ?

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SLIDE 13

Parametric cone

r h

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SLIDE 14

Parametric cylinder

r h

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SLIDE 15

Rendering faces: need location and normal

  • Need distance and orientation

relative to lights to compute reflected light

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SLIDE 16

Polygonal mesh

  • Simplest mesh: tetrahedron
  • Indexed mesh representation
  • Vertex list
  • Normal list
  • Face list
  • Non-indexed representation
  • List of faces with repeated vertices
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SLIDE 17

Polygonal mesh

  • Hill’s barn
  • 10 vertices
  • 7 faces
  • 7 normals
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SLIDE 18

File formats

  • STL
  • https://en.wikipedia.org/wiki/STL_(file_format)
  • OBJ
  • https://en.wikipedia.org/wiki/Wavefront_.obj_file
  • Many others