# L4: Matrices

Software and s/w Development

Dec 14, 2013 (4 years and 5 months ago)

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CLASS 4

CS770/870

Translation

Scale

Multiplying Matrices.

The R C rule

What happens when we do two translates?

What happens when we do two scales?

What happens when we translate and scale, or scale
then translate (Commutative?)

Exercise

Matrices in GL virtual machine

OpenGL maintains a matrix stack

glPushMatrix
() creates pushes a new 4x4 matrix on
the top of the stack containing a copy of the existing
top of stack matrix.

glTranslate
,
glScale

and
glRotate

cause a new matrix
to be
contructed

and post multiplied by the top of the
stack.

The matrix stack

();

M1

I

glRotatef
();

M1

IR1

glTranslatef
();

M1

IR1T1

glScalef
();

M1

IR1T1S1

glPushMatrix
();

M2

M1

glTranslatef
();

M2

M1T2

glPushMatrix
();

M3

M2

M1

M1

M1

M2

The view window to viewport transformation

In open GL

As a set of matrixes

Rotation about z axis

Derive rotation about z

Take notes

Rotations about x and y.

Exercise: do it by drawing

Given a square
glRectf
(
-
1,
-
1,1,1)

glRotatef
(
-
30.0,0,0,1);

glScalef
(2.0,1.0,1.0);

gLTranslatef
(2.0,0.0,0.0);

glRectf

(
-
1,
-
1,1,1);

glTranslatef
(0.0,2.0,0.0);

glScalef
(1.0,2.0,1.0);

glRotatef
(45,0.0,0.0, 1.0);

glRectf

(
-
1,
-
1,1,1);

The Scene window to viewport mapping

Scene window box (
Lsw
,
Rsw
,
Bsw
,
Tsw
)

Viewport box (
Lvp
,
Rvp
,
Bvp
,
Tvp
);

Exercise specify open
gl

commands to accomplish
this)

If time construct the matrix (Just the top row)

Some basic linear algebra (CH 4)

Dot and Cross Product.

A dot product of unit vectors gives the cosine of the
angle between two unit vectors

a
.
b

= (a
1
*b
1

+ a
2
*b
2

+ a
3
*b
3
)

= |a||
b|cos
(
q);

|a| =
Sqrt
(a
1
*a
1

+ a
2
*a
2

+ a
3
*a
3
);

a/|a| = a
1
/|a| + a
2/
|a| + a
3
/|a| // unit vector

Cross Product of two vectors

a
x

b

= (a
1
, a
2
, a
3
)
T
x

(b
1
, b
2
, b
3
)

The result is a vector

= (a
2

* b
3
)

(a
3

* b
2
),

(a
3

* b
1
)

(a
1

* b
3
),

(a
1

* b
2
)

(a
2

* b
1
)

If both vectors are unit vectors the result is at right
angles to the plane running through the other two.