Computer Graphic

Creator: Mohsen Asghari

Session 6

Fall 2014

Agenda Projection Matrix Perspective Projection Matrix Ortho Calculate Parameters of Perspective Model View matrix Camera (LookAt)

Projection Matrix (Perspective)

Projection Matrix (Perspective)

𝐴𝐵=𝑛𝐷𝐸=−𝑃 𝑧 𝐸𝐹=𝑃 𝑦

𝐵𝐶=𝑃 𝑠𝑦𝑃𝑠𝑦=

𝑃 𝑦∗𝑛−𝑃 𝑧

𝑃𝑠 𝑥=𝑃𝑥∗𝑛−𝑃 𝑧

Unit Cube

Projection Matrix (Perspective)𝑙≤ 𝑃𝑠𝑥 ≤𝑟 0≤𝑃 𝑠𝑥−𝑙≤𝑟 −𝑙

0≤𝑃𝑠𝑥− 𝑙𝑟 −𝑙  

≤1 0≤2∗𝑃𝑠𝑥−𝑙𝑟 − 𝑙  

≤2

−1≤2∗𝑃𝑠𝑥− 𝑙𝑟 −𝑙  

−1≤1

−1≤2∗𝑃𝑠𝑥

𝑟 −𝑙  −𝑟+ 𝑙𝑟 −𝑙

≤1

−1≤2𝑛∗𝑃𝑥

− 𝑃𝑧¿ ¿

𝑃𝑠 𝑥=𝑃𝑥∗𝑛−𝑃 𝑧

Projection Matrix (Perspective)

[ 2𝑛𝑟 − 𝑙 0𝑟+ 𝑙𝑟− 𝑙

0

… … … ……0

…0

…−1

…0

]−1≤

2𝑛∗𝑃𝑦

−𝑃𝑧 ¿¿ [2𝑛𝑟 − 𝑙

0𝑟+𝑙𝑟 − 𝑙

0

02𝑛𝑡−𝑏

𝑡+𝑏𝑡−𝑏

0

00

00

𝐴−1

𝐵0

]

Projection Matrix (Perspective)

𝑃𝑠𝑧=𝐴𝑃 𝑧+𝐵𝑃 𝑠𝑤=−𝑃 𝑧

−1=𝐴 𝑃𝑧+𝐵𝑃𝑠𝑤=−𝑃 𝑧

1=𝐴𝑃 𝑧+𝐵𝑃 𝑠𝑤=−𝑃 𝑧

𝑤h𝑒𝑛𝑃 𝑧=𝑛

𝑤h𝑒𝑛𝑃 𝑧= 𝑓{−𝑛𝐴+𝐵=−𝑛  − 𝑓𝐴+𝐵=− 𝑓  

Projection Matrix (Perspective)

A=−𝑓 +𝑛𝑓 −𝑛

B=−2 𝑓𝑛𝑓 −𝑛

[2𝑛𝑟 − 𝑙

0𝑟+𝑙𝑟 − 𝑙

0

02𝑛𝑡−𝑏

𝑡+𝑏𝑡−𝑏

0

00

00

−𝑓 +𝑛𝑓 −𝑛−1

−2 𝑓𝑛𝑓 −𝑛0

]

Question?All the matrix is related to Left = l right = rBottom = bTop = tNear = nFar = f

𝑀𝑎𝑡𝑟𝑖𝑥 4 .𝑪𝒓𝒆𝒂𝒕𝒆𝑷𝒆𝒓𝒔𝒑𝒆𝒄𝒕𝒊𝒗𝒆 (𝐹𝑂𝑉 ,𝐴𝑠𝑝𝑒𝑐𝑡 −𝑅𝑎𝑡𝑖𝑜𝑛 ,𝑁𝑒𝑎𝑟 ,𝐹𝑎𝑟 )

Calculate l,r,b,t BY (FOV) – (asp-r)

tan( 𝐹𝑂𝑉2 )∗𝑛𝑒𝑎𝑟=𝑡𝑜𝑝

𝑎𝑠𝑝𝑒𝑐𝑡 −𝑟𝑎𝑡𝑖𝑜𝑛=𝑤𝑖𝑑𝑡 hh𝑒𝑖𝑔h 𝑡

tan( 𝐹𝑂𝑉2 )∗𝑛𝑒𝑎𝑟 ∗𝑎𝑠𝑝−𝑟=𝑟𝑖𝑔h 𝑡

ProjectionMatrix Projection Matrix (GL_PROJECTION)

GL_PROJECTION matrix is used to define the frustum. This frustum determines which objects or portions of objects will be clipped out. Also, it determines how the 3D scene is projected onto the screen. (Please see more details

Ortho Perspective

Model View Matrix"World Matrix" and a "View Matrix", they are both transformation matrices.

View LookAt() Eye Target Up

END OF SESSION 5

ANY QUESTION?