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Transformations Computer Graphics Taku Komura

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Page 1: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

TransformationsComputer Graphics

Taku Komura

Page 2: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Setting Objects in the Scene

● Once the models are prepared, we need to place them in the environment

● We need to know the vertex locations of the objects in the world coordinate system

● But objects are only defined in their own local coordinate system

Page 3: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Transformations

We translate, rotate and scale the vertices in the world coordinate system

Page 4: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Overview

•Homogeneous transformation•Background•Conversion of coordinates•2D, 3D cases

•Camera transformation

Page 5: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Homogenous Transformation

•Background•Conversion of coordinates•2D, 3D cases

Page 6: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Background of Homogeneous Transformation

Translation

Page 7: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Background of Homogeneous Scaling

Page 8: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Background of Homogeneous

Page 9: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Background of Homogeneous

Page 10: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 11: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Subsequently applying transformations?

•We can apply the above mentioned transformations to translate, scale and rotate the object vertices•What if we want a combination of such operations?

Page 12: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Subsequently applying transformations

Page 13: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Subsequently applying transformations(2)

•Matrix operations are not commutative•AB ≠ BA

Page 14: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 15: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 16: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 17: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Rotation, Translation

The matrix of rotation and then translation is often used so it may be worth to remember it.

Page 18: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Transformations between different coordinate systems

We can interpret that the transformation matrix is converting the location of vertices between different coordinate systems

Page 19: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Example

Page 20: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Xl viewed from g

Xg viewed from l

Page 21: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Transformations between different coordinate systems (2)

We can forget about local-global and generalize the idea

Page 22: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Sequential Transformations between different coordinate systems

We have a series of coordinate systems, a,b,c,d.•We only know the relative transformations between the adjacent ones:

••What is the position of a point vd defined in coordinate d, when viewed from coordinate a?

Page 23: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

What is the position of Vd with respect to c?

Page 24: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

What is the position of Vd with respect to c?

Page 25: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

What is the position of Vd with respect to b?

Page 26: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

What is the position of Vd with respect to b?

Page 27: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

What is the position of Vd with respect to a?

Page 28: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Transformations of coordinate

A series of transformation matrices multiplied from left to right

Page 29: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Multiplying from the left and right

1

2

3

1

2

3

Page 30: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Why is this useful?

Car example:–Starting the car from one location, we do a series of operations (driving forward, turning)- What is the final location of the car?Robot Example- The body size of the robots are known and the robot bends it joints- What is the position of its hand?

Page 31: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Quiz

I sat in the car, and find the side mirror is 0.4m on my right •I started my car and drove 5m forward, turned 30 degrees to right, moved 5m forward again•What is the position of the side mirror now, relative to where I was sitting in the beginning?

Page 32: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Hint

The matrix of first driving forward 5m is

The matrix to turn to the right 30degrees (rotating -30 around the origin) is

Page 33: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Solution

The local-to-global transformation matrix at thelast configuration of the car is

Page 34: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 35: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 36: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 37: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 38: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 39: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 40: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex
Page 41: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Rotation and Translation in 3D

Page 42: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Viewing

•Now we have the world coordinates of all the vertices•Now we want to convert the scene so that it appears in front of the camera

Page 43: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

View Transformation

● We want to know the positions in the camera coordinate system

● We can compute the camera-to-world transformation matrix using the orientation and translation of the camera from the origin of the world coordinate system

Page 44: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

View Transformation

We want to know the positions in the camera coordinate system vg = Mg←c vc

Point in the world coordinate Point in thecamera coordinate

Camera-to-worldtransformation

vc = Mg←c vg = Mc←g vg

-1

Page 45: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Summary

•Transformations: translation, rotation and scaling•Using homogeneous transformation, 2D (3D) transformations can be represented by multiplication of a 3x3 (4x4) matrix•Multiplication from left-to-right can be considered as the transformation of the coordinate system•Reading: Shirley et al. Chapter6 Foley et al. Chapter 5, Appendix 2 sections A1 to A5 for revision and further background (Chapter 5)

Page 46: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Lab session in the next lecture slot

Appleton Tower 5.05 from 11am-12pm on Friday 27th September

About using OpenGL and some description about the assignment

Page 47: Taku Komura Computer Graphics - School of Informatics · Setting Objects in the Scene Once the models are prepared, we need to place them in the environment We need to know the vertex

Modern Textbook

Shirley et al.


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