Graphics - Basic Transforms

translation, scaling ,shearing, rotation

what are the 4 fundamental transforms in computer graphics

translation

only moves the matrix, does not alter the size or orientation of it

2D matrix is

x+tx

y+ty

\
homogenous matrix is

1 0 tx

0 1 ty

0 0 1

scaling

used to stretch or shrink a points location along the x or y axis or along both axes at the same time

2D matrix is

sx 0

0 sy

\
homogenous matrix is

sx 0 0

0 sy 0

0 0 1

uniform

a scaling matrix is ______ when both dimensions are the same (non-(answer)) when the dimensions are different

<

a scaling factor of <\\> 0 will result in a reflection about the corresponding axis

shrink

when the scaling factor is 0<x<1, the matrix will _____ in the corresponding direction

collapse

a scaling factor of 0 will _____ the matrix to that vector (make it lie on the axis a.k.a become a vector)

shearing

used to proportionately change the values along one axis based on the distance along the other axis

2D matrix is

1 hx

hy 1

\
homogenous matrix is

1 hx 0

hy 1 0

0 0 1

X

shifting in the ___ direction results in the X values getting larger as y value increases

Y

shifting in the ____ direction results in the Y values getting larger as x value increases

origin

rotation is always with respect to the

rotation

turning the matrix about a point

2D matrix is

cosθ -sinθ

sinθ cosθ

\
homogenous matrix is

cosθ -sinθ 0

sinθ cosθ 0

0 0 1

no

do multiple transformations multiplied in different orders yield the same result

cannot

we (can/cannot) rotation and translation transforms

T

T/F scaling, shearing, and rotation can all be combined into a single transformation matrix

associative

combining matrix operations is

addition

translation cannot be combined with other transform matrices bc translation uses

homogenous

being able to represent translation, scaling, rotation, and shearing in a uniform way is a benefit of ________ coordinate matrix

we also get to combine them using this standard

\[0 0 1\]

the third column of rotation, shearing, and scaling homogenous matrices is _____ because they do not translate

\[0 0 1\]

the third row for all 4 transform matrices in homogeneous coordinates

this is because they are all affine transformations

w

points in homogenous coordinates have an extra value called

1

w = ___ for points

0

w = ____ for vectors

polar

rotation is much easier if you express coordinates in _____ form instead of cartesian form

affine

an ______ transformation is any transformation that preserves points-on-a-line and proportionality of distance

\
a point stays the same

\
a line or line segment is still the same (does not get compressed into a point)

\
two lines or line segments that were parallel are still parallel

\
proportional positions of points on a line or line segment remain the same

proportionality

after the transformation, even of the length of a line segment may change, the proportion of distances of points on that line segment remain constant

rigid

a ____ body is one that we are not allowed to deform

rigid

translation, rotation, and scaling, when using a scale of 1 or -1, are both (rigid/nonrigid) transformations

never

shearing is (always/never) a rigid body transformation

translation

Homogenous _______ 3D matrix

1 0 0 tx

0 1 0 ty

0 0 1 tz

0 0 0 1

scaling

Homogenous ________ 3D matrix

sx 0 0 0

0 sy 0 0

0 0 sz 0

0 0 0 1

shearing

homogenous _____ 3D matrix

1 hyx hzx 0

hxy 1 hxy 0

hxz hyz 1 0

0 0 0 1

rotation, x

homogenous ___ __about__ ___ axis 3D matrix

1 0 0 0

0 cosθx -sinθx 0

0 sinθx cosθx 0

0 0 0 1

rotation, y

homogenous ___ __about__ ___ axis 3D matrix

cosθy 0 sinθy 0

0 1 0 0

\-sinθy 0 cosθy 0

0 0 0 1

rotation, z

homogenous ___ __about__ ___ axis 3D matrix

cosθz -sinθz 0 0

sinθz cosθz 0 0

0 0 1 0

0 0 0 1