CPSC 453 Tutorials

Xin LiuOct 16, 2013

HW1 review

• Why I was wrong?

Q1Determine the affine transformation on the plane taking the triangle with vertices (1, 1)(1, 2), and (3, 3) to the equilateral triangle with with vertices (1, 0), (-1, 0), and (0, sqrt(3))

Solution:

1 1 01 1 31 2 3 0 0 31 1 1 1 1 1

T

11 1 0 3/ 2 2 3/ 21 1 30 0 3 1 2 3 3 / 2 0 3 / 21 1 1 1 1 1 0 0 1

T

Refer to a Linear Algebra textbook for Inverse Matrix calculation

Q2Let P, Q, R be points on the 2D affine plane. Show that for an arbitrary scalar, is a point but is a vector

22 2 1 1X u P u u Q u R

4 3Y P Q R

Solution: for any point

The last component of X is 1, becauseTherefore, X is a point.

The last component of Y is 0, becauseTherefore, Y is a vector.

1

P

P

xP y

1

Q

Q

xQ y

1

R

R

xR y

22 1 2 1 1 1 1 1u u u u u

4 1 1 1 3 1 0

Q3Define what it means for a transformation in Rn to preserve angles.(a) Show that an isometry preserves angles.(b) Give an example of a transformation that preserves angles but is not an isometry.

Solution:Let T be a linear transformation in Rn

. T is angle preserving iff

An isometry transform , L(u) is an orthogonal transformation

, ,, n

u v TuTvu v

u v Tu Tv

T u L u t

T

T u T U O T U T O L U t L O t L u

, , ,TuTv Lu Lv u v

Q4Let the frame F on the plane be obtained from the cartesian reference frame by acounter-clockwise rotation about the origin through 135 degrees. Find the transfermatrices. An ellipse has equation 5x2+6xy+ty2=1 in cartesian; what is its equationin the frame F?

cos135 sin135 1 11sin135 cos135 1 12

FEA

''

FE

x xAy y

1 ' '21 ' '2

x x y

y x y

2 22 8 1x y

2 21 1 1 15 ' ' 6 ' ' ' ' ' ' 1

2 2 2 2x y x y x y x y

Q5Find the transformation matrix for a rotation by a 120-degree angle about the axis definedBy the unit vector r = 1/sqrt(3)(1, 1, 1). (This of course can be done using the resultof the previous exercise, but you might be able to guess the matrix directly by consideringwhat the transformation does to the unit cube [0, 1]3.

A permutation of axis: x->y, y->z, z->x

0 0 11 0 00 1 0