Transformations. Lars Vidar Magnusson. August 24,

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1 Transformations Lars Vidar Magnusson August 24,

2 2D Translation To translate an object is to move it in two-dimensinal space. If we have a point p 1 defined by a vector [x 1, y 1 ], we can easily define a second point p 2 by starting with the components of p 1 and adding offset. x 2 = x 1 + x t y 2 = y 1 + y t This can be written in vector form as: p 2 = [x 1 + x t, y 1 + y t ] = p 1 + t

3 2D Scaling To scale an object is to resize it in two-dimensional space. Again we start with a point p 1 defined by a vector [x 1, y 1 ]. This point can be resized into a new point p 2 by multiplying each component with a scale factor. x 2 = x 1 x s y 2 = y 1 y s In vector form this can be written as: p 2 = [x 1 x s, y 1 y s ] Note that we are scaling the point in relation to the origin.

4 2D Rotation Rotation, like with scaling, is done relative to the origin of the two-dimensional world, but unlike the two other transformations, rotation is not as easy to deduce. We have to start by defining the points p 1 and p 2. p 1 = [x 1, y 1 ] = [r cos α, r sin α] p 2 = [x 2, y 2 ] = [r cos(α + β), r sin(α + β)] To make any further progress, it might be helpful to refresh som simple trigonometry. cos(α + β) = cos α cos β sin α sin β sin(α + β) = cos α sin β + sin α cos β

5 2D Rotation Continued We can insert the sum of angles formula and then substitue the expression for p 1 in the expression for p 2. p 2 = [r cos α cos β r sin α sin β, r cos α sin β + r sin α cos β] = [x 1 cos β y 1 sin β, x 1 sin β + y 1 cos β] This results in the following component expressions for p 2 : x 2 = x 1 cos β y 1 sin β y 2 = x 1 sin β + y 1 cos β

6 Three-dimensional Worlds Three-dimensional worlds, allthough they have only one more dimension, becomes much more complex than two dimensional worlds. One of the reasons for this is that it is not definite how to orient the world. We can use either left-handed or right-handed orientation. OpenGL use right-handed orientation by default, and so do the lectures notes.

7 3D Translation Translation in three dimensions is not that different from translation in two dimensions. We just extend the expressions with a third value for the z axis. x 2 = x 1 + x t y 2 = y 1 + y t z 2 = z 1 + z t This can be written in vector form as: p 2 = [x 1 + x t, y 1 + y t, z 1 + z t ] = p 1 + t

8 3D Scaling Three-dimensional scaling, like three-dimensional translation, can be expressed by extending the expressions deduced for two-dimensional scaling with a third value representing the z axis. x 2 = x 1 x s y 2 = y 1 y s z 2 = z 1 z s In vector form this can be written as: p 2 = [x 1 x s, y 1 y s, z 1 z s ]

9 3D Rotation Three-dimensional rotation is a bit more complex than the two others in three dimensions. The reason for this is that we go from one possible rotation axis in two dimensions to three. The easy extension leads to rotation around the z axis. x 2 = x 1 cos β y 1 sin β y 2 = x 1 sin β + y 1 cos β z 2 = z 1 This can be written in vector form as: p 2 = [x 1 cos β y 1 sin β, x 1 sin β + y 1 cos β, z 1 ]

10 3D Rotation Continued Rotation around the x axis can be expressed by: x 2 = x 1 y 2 = y 1 cos β z 1 sin β z 2 = y 1 sin β + z 1 cos β And finally, around the y axis can be expressed by: x 2 = x 1 cos β + z 1 sin β y 2 = y 1 z 2 = x 1 sin β + z 1 cos β Note that the rotations changes if the coordinate system is left-handed.

11 Matrices All the deduced equation sets for the different types of transformation are linear, which allows us to write them as matrices Since we have three dimensions, the intuitive approach would be to use 3x3 matrices to represent the transformations, but this is not a good solution.

12 Homogeneous Transformation Matrices Homogeneous transformation matrices make it possible to represent translation as well as scaling and rotation. m m 11 m 12 m 11 m 12 m m 21 m 22 m 23 m 21 m 22 m 23 0 m m 31 m 32 m 31 m 32 m Multiplication with a vector v would then be set up like this: m 11 m 12 m 13 0 m 21 m 22 m 23 0 m 31 m 32 m 33 0 v x v y v z 1 = v x v y v z 1

13 Translation and Scaling Matrices Due to the extra dimension, we can insert the translation into the columns which will be multiplied with new 1 in the vectors t x M t = t y t z Scaling can be written in matrix form by inserting the scaling factors into the diagonal. s x M s = 0 s y s z 0

14 Rotation Matrices The three rotation matrices around the x, y and z can be written respectively in matrix form as: M rx = 0 cos α sin α 0 0 sin α cos α 0 cos α 0 sin α 0 M ry = sin α 0 cos α 0 cos α sin α 0 0 M rz = sin α cos α

15 Identity Matrix The identity matrix, typically denoted by I, is a common matrix type I = Multiplying any matrix M or vector v with I and you get the same result, which makes it equivalent to multiplying with 1.

16 Mirror Matrix The mirror matrix flips one of the axis in the coordinate system to achieve mirrored geometry M my = The matrix shown mirrors the y axis.

17 Shear Matrix A Shear matrix will distort the geometry by adding scale component outside of the matrix diagonal M my = The matrix shown is set up so that the x axis is dependent on y.

18 Projection Matrix Matrices are used to for both orthogonal and perspective projection as well as standard transformation. The following matrix projects the world onto the xy plane M oz = An easy perspective projection might look like this: M pz =

19 Transformations in OpenGL Transformations in OpenGL is handled by the following functions. g l L o a d I d e n t i t y ( ) ; g l T r a n s l a t e f ( tx, ty, t z ) ; g l S c a l e f ( sx, sy, s z ) ; g l R o t a t e f ( angle, rx, ry, r z ) ; g l M u l t M a t r i x f ( m a t r i x ) ; The f postfix specifies the type of the parameters. Since there is no function overloading in C, each function has several different versions for different types e.g. gltranslated.