CS 6958 LECTURE 7 TRIANGLES, BVH January 29, 2014 Triangles 2 - - PowerPoint PPT Presentation

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CS 6958 LECTURE 7 TRIANGLES, BVH January 29, 2014 Triangles 2 - - PowerPoint PPT Presentation

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CS 6958 LECTURE 7 TRIANGLES, BVH January 29, 2014 Triangles 2 Lets try to derive an intersection test Barycentric Coordinates 3 1 2 3 0 1 , 2 , 3 1 1 1 + 2 + 3 = 1 2

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SLIDE 1

CS 6958 LECTURE 7 TRIANGLES, BVH

January 29, 2014

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SLIDE 2

Triangles

 Let’s try to derive an intersection test

2

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SLIDE 3

Barycentric Coordinates

3

0 ≤ 𝑐1, 𝑐2, 𝑐3 ≤ 1 𝑐1 + 𝑐2 + 𝑐3 = 1 𝑄 = 𝑐1𝑄

1 + 𝑐2𝑄2 + 𝑐3𝑄3

𝑄 = 𝑐1𝑄

1 + 𝑐2𝑄2 + 1 − 𝑐1 − 𝑐2 𝑄3

𝑄2 𝑄3 𝑄

1

𝑄 𝑐3 𝑐2 𝑐1

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SLIDE 4

Barycentric Coordinates

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0 ≤ 𝑐1, 𝑐2, 𝑐3 ≤ 1 𝑐1 + 𝑐2 + 𝑐3 = 1 𝑄 = 𝑐1𝑄

1 + 𝑐2𝑄2 + 𝑐3𝑄3

𝑄 = 𝑐1𝑄

1 + 𝑐2𝑄2 + 1 − 𝑐1 − 𝑐2 𝑄3

𝑃 + 𝑢𝑊 = 𝑐1𝑄

1 + 𝑐2𝑄2 + 1 − 𝑐1 − 𝑐2 𝑄3

𝑄2 𝑄3 𝑄

1

𝑄 𝑐3 𝑐2 𝑐1

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SLIDE 5

𝑃 + 𝑢𝑊 = 𝑐1𝑄

1 + 𝑐2𝑄2 + 1 − 𝑐1 − 𝑐2 𝑄3

−𝑢𝑊 + 𝑐1 𝑄

1 − 𝑄3 + 𝑐2 𝑄2 − 𝑄3 = 𝑃 − 𝑄3

𝑓1 = 𝑄

1 − 𝑄3

𝑓2 = 𝑄2 − 𝑄3 𝑡 = 𝑃 − 𝑄3 −𝑊

𝑦

𝑓1𝑦 𝑓2𝑦 −𝑊

𝑧

𝑓1𝑧 𝑓2𝑧 −𝑊

𝑨

𝑓1𝑨 𝑓2𝑨 𝑢 𝑐1 𝑐2 = 𝑡𝑦 𝑡𝑧 𝑡𝑨

Barycentric Coordinates

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𝑄2 𝑄3 𝑄

1

𝑄 𝑐3 𝑐2 𝑐1

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SLIDE 6

𝑢 =

𝑡𝑦 𝑓1𝑦 𝑓2𝑦 𝑡𝑧 𝑓1𝑧 𝑓2𝑧 𝑡𝑨 𝑓1𝑨 𝑓2𝑨 −𝑊

𝑦

𝑓1𝑦 𝑓2𝑦 −𝑊

𝑧

𝑓1𝑧 𝑓2𝑧 −𝑊

𝑨

𝑓1𝑨 𝑓2𝑨

, 𝑐1 =

−𝑊

𝑦

𝑡𝑦 𝑓2𝑦 −𝑊

𝑧

𝑡𝑧 𝑓2𝑧 −𝑊

𝑨

𝑡𝑨 𝑓2𝑨 −𝑊

𝑦

𝑓1𝑦 𝑓2𝑦 −𝑊

𝑧

𝑓1𝑧 𝑓2𝑧 −𝑊

𝑨

𝑓1𝑨 𝑓2𝑨

,𝑐2 =

−𝑊

𝑦

𝑓1𝑦 𝑡𝑦 −𝑊

𝑧

𝑓1𝑧 𝑡𝑧 −𝑊

𝑨

𝑓1𝑨 𝑡𝑨 −𝑊

𝑦

𝑓1𝑦 𝑓2𝑦 −𝑊

𝑧

𝑓1𝑧 𝑓2𝑧 −𝑊

𝑨

𝑓1𝑨 𝑓2𝑨

 In reality too slow for intersections, but we can

do better!

Solution – Cramer’s Rule

6

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SLIDE 7

Scalar Triple Product

 Scalar triple product

𝐵 ∙ 𝐶 × 𝐷 = 𝐶 ∙ 𝐷 × 𝐵 = 𝐷 ∙ 𝐵 × 𝐶

 Also expressed as a determinant

𝐵𝑦 𝐵𝑧 𝐵𝑨 𝐶𝑦 𝐶𝑧 𝐶𝑨 𝐷𝑦 𝐷𝑧 𝐷𝑨

 This comes in handy

𝐵𝑈 = 𝐵

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SLIDE 8

Faster Solution

𝑒𝑓𝑜𝑝𝑛 = −𝑊

𝑦

−𝑊

𝑦

−𝑊

𝑦

𝑓1𝑦 𝑓1𝑧 𝑓1𝑨 𝑓2𝑦 𝑓2𝑧 𝑓2𝑨 = −𝑊 ∙ 𝑓1 × 𝑓2 = −𝑓1 ∙ 𝑓2 × 𝑊 = 𝑓1 ∙ 𝑊 × 𝑓2 𝑒𝑓𝑜𝑝𝑛 = 𝑓1 ∙ 𝑊 × 𝑓2 𝑢 = 𝑓2 ∙ 𝑡 × 𝑓1 𝑒𝑓𝑜𝑝𝑛 𝑐1 = 𝑡 ∙ 𝑊 × 𝑓2 𝑒𝑓𝑜𝑝𝑛 𝑐1 = 𝑊 ∙ 𝑡 × 𝑓1 𝑒𝑓𝑜𝑝𝑛

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SLIDE 9

Ray-Triangle Intersection

𝑓1 = 𝑄

1 − 𝑄3

𝑓2 = 𝑄2 − 𝑄3 𝑠

1 = 𝑊 × 𝑓2

𝑒𝑓𝑜𝑝𝑛 = 𝑓1 ∙ 𝑠

1

if(𝑏𝑐𝑡 𝑒𝑓𝑜𝑝𝑛 < 𝜗) miss, return;

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SLIDE 10

Ray-Triangle Intersection

𝑓1 = 𝑄

1 − 𝑄3

𝑓2 = 𝑄2 − 𝑄3 𝑠

1 = 𝑊 × 𝑓2

𝑒𝑓𝑜𝑝𝑛 = 𝑓1 ∙ 𝑠

1

if(𝑏𝑐𝑡 𝑒𝑓𝑜𝑝𝑛 < 𝜗) miss, return; 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 = 1 𝑒𝑓𝑜𝑝𝑛 s = 𝑃 − 𝑄3 𝑐1 = 𝑡 ∙ 𝑠

1 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛

if(𝑐1 < 0 ∥ 𝑐1 > 1) miss, return;

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SLIDE 11

Ray-Triangle Intersection

𝑓1 = 𝑄

1 − 𝑄3

𝑓2 = 𝑄2 − 𝑄3 𝑠

1 = 𝑊 × 𝑓2

𝑒𝑓𝑜𝑝𝑛 = 𝑓1 ∙ 𝑠

1

if(𝑏𝑐𝑡 𝑒𝑓𝑜𝑝𝑛 < 𝜗) miss, return; 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 = 1 𝑒𝑓𝑜𝑝𝑛 s = 𝑃 − 𝑄3 𝑐1 = 𝑡 ∙ 𝑠

1 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛

if(𝑐1 < 0 ∥ 𝑐1 > 1) miss, return; 𝑠2 = 𝑡 × 𝑓1 𝑐2 = 𝑊 ∙ 𝑠2 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 if(𝑐2 < 0 ∥ 𝑐1 + 𝑐2 > 1) miss, return; t = 𝑓2 ∙ 𝑠2 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 hit! save 𝑐1 and 𝑐2 for interpolation

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SLIDE 12

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Operation add / sub / mult compare divide 𝑓1 = 𝑄

1 − 𝑄3

3 𝑓2 = 𝑄2 − 𝑄3 3 𝑠

1 = 𝑊 × 𝑓2

9 𝑒𝑓𝑜𝑝𝑛 = 𝑓1 ∙ 𝑠

1

5 if(𝑏𝑐𝑡 𝑒𝑓𝑜𝑝𝑛 < 𝜗) miss, return; 2 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 = 1 𝑒𝑓𝑜𝑝𝑛 1 s = 𝑃 − 𝑄3 3 𝑐1 = 𝑡 ∙ 𝑠

1 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛

6 if(𝑐1 < 0 ∥ 𝑐1 > 1) miss, return; 2 𝑠

2 = 𝑡 × 𝑓1

9 𝑐2 = 𝑊 ∙ 𝑠2 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 6 if(𝑐2 < 0 ∥ 𝑐1 + 𝑐2 > 1) miss, return; 1 2 t = 𝑓2 ∙ 𝑠2 𝑗𝑜𝑤𝐸𝑓𝑜𝑝𝑛 6 hit! save 𝑐1 and 𝑐2 for interpolation 2 total 20 / 29 / 45 / 51 2 / 4 / 6 / 8 0 / 1 / 1 / 1

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SLIDE 13

Normals

 Flat shaded triangles

𝑂 = 𝑓1 × 𝑓2

 Smooth shaded (per-vertex normals)

𝑂 = 𝑐1𝑂1 + 𝑐2𝑂2 + 1 − 𝑐1 − 𝑐2 𝑂3

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SLIDE 14

Updates to Hitrecord

 include barycentric coordinates  include computed normal

 alternatively, save per-vertex normals and

interpolate before shading

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SLIDE 15

Which Operation Most Costly?

foreach frame foreach pixel foreach sample generate ray intersect ray with objects shade intersection point

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SLIDE 16

End

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