TNM046: Datorgrafik Triangle and Normal

TNM046: Datorgrafik Triangle and Normal

Sasan Gooran

VT 2013

Vertex and Triangle

Vertex: ( x y z )

Triangle: array of Vertex (vertices)

3D object: array of Triangle (triangles)

Example: Triangle[1] : (10,10,10), (10,20,10), (10,20,20) Triangle[2] : (10,10,10), (10,20,10), (10,10,20)

Not a good approach

3

Vertex and Triangle

Vertex: ( x y z )

Vertex table: array of Vertex

Triangle table: array of integers

Example: Vertex table: (10,10,10) , (10,20,10) , (10,20,20), (10,10,20), …..

Triangle table: (1,2,3) , (1,2,4) , ….

3D object: Vertex table + Triangle table

Now all vertices in the object is in one list (table).

Vertex and Triangle

Example: Present the following object using vertex and

triangle table. The distance from P₁, P₂ and P₃ to the origin (P₀) is unity (1).

y z

p₀ p₁

p₂ p₃

Triangle 1:

p₀p₁p₃

Triangle 2:

p₀p₂p₃

Triangle 3:

p₁p₂p₃

Vertex table:

p₀: (0,0,0) p₁: (1,0,0) p₂: (0,1,0) p₃: (0,0,1) Answer:

Triangle table:

Triangle 0: (0,1,2) Triangle 1: (0,1,3)

5

Vertex and Triangle

Assume triangle 1 (p₀p₁p₃).

p₀ = (0,0,0), p₁ = (1,0,0), p₃ = (0,0,1),

x z

p₀ p₁

p₃

d 

d 

d 

Difference vectors:

d 

= (1, 0, 0) − (0, 0, 0) = (1, 0, 0)

d 

= (0, 0,1) − (1, 0, 0) = (−1, 0,1)

d 

= (0, 0, 0) − (0, 0,1) = (0, 0, −1)

Triangle and Normal

x z

p₀ p₁

p₃

d 

d 

d 

y z

p₀ p

p₂ p₃

Triangle 1:

p₀p₁p₃

d 

× 

d

= (1, 0, 0) × (−1, 0,1) =

ˆx ˆy ˆz

1 0 0

−1 0 1

= (0, −1, 0)

d 

× 

d

= (−1, 0,1) × (0, 0, −1) = (0, −1, 0) d 

× 

d

= (0, 0, −1) × (1, 0, 0) = (0, −1, 0)

7

Triangle and Normal

x z

p₀ p₁

p₃

d 

d 

d 

Triangle 1:

p₀p₁p₃

N

= 

d

× 

d

= 

d

× 

d

= 

d

×  d

Three possible starting points For this triangle three lists are possible:

(0,1,3) or (1,3,0) or (3,0,1)

In order to calculate the normal pointing outwards:

Find the vector from the first point to the second (call A₁) Find the vector from the second point to the third (call A₂)

N

= 

A

× 

A

points outwards

Triangle and Normal

Triangle table: Let your thumb of the right hand stick out of the object. Pick the first point, then the sense of your fingers gives you the other two points in the right sequence.

y z

p₀ p₁

p₂ p₃

Triangle 1:

p₀p₁p₃

Triangle 2:

p₀p₂p₃

Triangle 3:

p₁p₂p₃

Triangle table:

Triangle 0: (0,2,1) Triangle 1: (0,1,3) Triangle 2: (0,3,2) Triangle 3: (1,2,3)

Compare this triangle table with the one on page 4.

9

Triangle and Normal

Therefore the following 3D object can be defined as:

x

y z

p₀ p₁

p₂ p₃

Triangle table:

(0,2,1) (0,1,3) (0,3,2) (1,2,3)

Vertex table:

(0,0,0) (1,0,0) (0,1,0) (0,0,1)