# transformations: Affine planar transformation matrix In RConics: Computations on Conics

## Description

(3 \times 3) affine planar transformation matrix corresponding to reflection, rotation, scaling and translation in projective geometry. To transform a point p multiply the transformation matrix A with the homogeneous coordinates (x,y,z) of p (e.g. p_{transformed} = Ap).

## Usage

 1 2 3 4 reflection(alpha) rotation(theta, pt=NULL) scaling(s) translation(v) 

## Arguments

 alpha the angle made by the line of reflection (in radian). theta the angle of the rotation (in radian). pt the homogeneous coordinates of the rotation center (optional). s the (2 \times 1) scaling vector in direction x and y. v the (2 \times 1) translation vector in direction x and y.

## Value

A (3 \times 3) affine transformation matrix.

## References

Richter-Gebert, J<c3><bc>rgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Real and Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 p1 <- c(2,5,1) # homogeneous coordinate # rotation r_p1 <- rotation(4.5) %*% p1 # rotation centered in (3,1) rt_p1 <- rotation(4.5, pt=c(3,1,1)) %*% p1 # translation t_p1 <- translation(c(2,-4)) %*% p1 # scaling s_p1 <- scaling(c(-3,1)) %*% p1 # plot plot(t(p1),xlab="x",ylab="y", xlim=c(-5,5),ylim=c(-5,5),asp=1) abline(v=0,h=0, col="grey",lty=1) abline(v=3,h=1, col="grey",lty=3) points(3,1,pch=4) points(t(r_p1),col="red",pch=20) points(t(rt_p1),col="blue",pch=20) points(t(t_p1),col="green",pch=20) points(t(s_p1),col="black",pch=20) 

RConics documentation built on May 30, 2017, 5:22 a.m.