TNM046: Datorgrafik Triangle and Normal

TNM046: Datorgrafik Triangle and Normal

Sasan Gooran

VT 2014

Computer Graphics

What?

“Almost everything on computers that is not text or sound “ The representation and manipulation of image data by

computer

Techniques to create and manipulate images

Where?

Television, Newspapers (weather reports), Medical investigations and surgical procedures

3

Computer Graphics

Applications:

Entertainments (Movies, Video games) Computer Aided Design (CAD)

Medical Applications Scientific Visualization Virtual Reality

….

Computer Graphics

5

Computer Graphics, Objects

In order to define an object, you need to first specify the

coordinates of the vertices. Then you define the triangles the object consists of. Each triangle is defined by specifying its three vertices.

In order to perform different transformations (e.g. scaling, translation, rotation etc.) on an object you transform the

object’s vertices. This is done by multiplying the appropriate transformation matrix with the coordinates of the vertices.

Computer Graphics, TNM046

Triangle and Normal

2D and 3D Transformations Light Sources and Shading Perspective Projection

Culling and Clipping Texture Mapping

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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

Every vertex is repeated

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).

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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).

x

y z

p₀ p₁

p₂ p₃

Triangle 0: p₀p₁p₂ 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) Triangle 2: (0,2,3) Triangle 3: (1,2,3)

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)

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Triangle and Normal

x z

p₀ p₁

p₃

d 

d 

d 

x

y z

p₀ p₁

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)

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)

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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.

x

y z

p₀ p₁

p₂ p₃

Triangle 0: p₀p₁p₂ 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.

Triangle and Normal

Therefore the following 3D object can be defined as:

y z

p₀ p₁

p₂ p₃

Triangle table:

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

Vertex table:

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

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Triangle and Normal

In OpenGL, together with the vertex coordinate you also specify the normal vector and the texture coordinates (for texture mapping). Texture mapping will be discussed later.

x

y z

p₀ p₁

p₂ p₃

Let us know look at vertex 0 (p₀)

Vertex 0 is the junction oft hree triangles,

each of which has its own normal vector, i.e., (0,0,-1), (0, -1, 0) and (-1, 0, 0). Therefore vertex 0 is defined three times:

Vertex p₀:

Nr coordinate normal 0: (0,0,0) (0, 0, -1) 1: (0,0,0) (0, -1, 0) 2: (0,0,0) (-1, 0, 0)

Triangle and Normal

y z

p₀ p₁

p₂ p₃

Vertex table:

Nr coordinate normal 0: (0,0,0) (0, 0, -1) 1: (0,0,0) (0, -1, 0) 2: (0,0,0) (-1, 0, 0) 3: (1,0,0) (0, 0, -1) 4: (1,0,0) (0, -1, 0)

5: (1,0,0) (0.577,0.577,0.577) 6: Fill the rest, will be discussed in the class (första lektionen)

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Triangle and Normal

x

y z

p₀ p₁

p₂

p₃ Triangle table:

Nr

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

Fill the rest, will be discussed in the class (första lektionen)

Triangle 0, consists of (p₀, p₂, p₁). For p₀ we now have

three vertices, 0, 1 and 2. We choose the one that has the normal associated with Triangle 0, which will be vertex 0.

For p₁ we should choose between vertex nr 3, 4 and 5.

The correct one in this case is vertex nr 3 because it has the normal vector associated with Triangle 0.