Exploration Discussion

 

Overview

The Exploration materials and Exploration Discussion provide students with practical motivation, and insight into common application areas.

Instructions

Within the Module 2 Exploration, after reading through the Exploration tabs, and checking your knowledge in the Check  Understanding tab, click on the Discuss! tab.

Choose one of the applications, and one of the numbered Discussion Topics for that application to explore with your classmates.

Steps for Exploration Discussion Posts

1. Create a thread that answers the question(s) posed in the Discussion Topic to the best of your ability. Provide as much detail as possible.
2. Title your thread with the name of the application and the number of the Discussion Topic (e.g. Computer Graphics, Discussion Topic #2), and restate the Discussion Topic at the top of your post.
3. Take part in the resulting discussions for this Discussion Topic, and at least one other Discussion Topic from your chosen Application, or from one of the other Applications. You must have at least one main post, and at least two other substantive discussion posts to satisfy the requirements for this Exploration Discussion.

See the Schedule in the Syllabus and Schedule Module for due dates and the Rubric attached to this Discussion for grading information.

EXPLORATION OF MATRIX ALGEBRA AND

DETERMINANTS

Introduction Computer Graphics Areas and Volumes Check Understanding
Discuss! References

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Note: If equations don’t render correctly when initially viewed, refresh your
browser page and wait for it to load completely before visiting any of the
tabs.

Linear transformations play an important role in computer graphics,
especially in the animation of a figure in a two- or three-dimensional
coordinate system. Here, we explore the two-dimensional case, in which we
have an isosceles triangle representing an arrowhead described by the vector

coordinates, .

The top of this triangle represents the point of the arrowhead, currentl

y

pointing straight upward as shown in Figure 2.1.1.

[ ],  [ ],  and [ ]
−

1

−1

0
0

1
−1

Listen

https://app.readspeaker.com/cgi-bin/rsent?customerid=12160&url=https%3A%2F%2Fmycourses.cccs.edu%2Fcontent%2Fenforced%2F356437-S_CCA_MAT2540X00_202520%2FModule%252002%2FM2_Exploration.html&lang=en_us&readid=d2l_read_element_1

Fig. 2.1.1 (generated from): Wolfram Alpha: Computational Knowledge Engine. (2009). Retrieved
March 26, 2016, from https://www.wolframalpha.com

Figure 2.1.1: Graph of arrowhead described by the vector coordinates,

The arrowhead can be represented by a matrix with these vectors as its
columns,

The order of the columns is not important, and the vertices are all that is
needed to reconstruct the arrowhead. We will move the arrowhead around
by performing linear transformations on its vertices. Properties of linear
transformations tell us that lines between the original vertices will be
transformed to lines between the resulting vertices, so our transformed
object will still be an arrowhead (possibly stretched, projected, or otherwise
moved around). This means that we only need to know what a linear
transformation does to the vertices in the matrix A to know what it does to
the whole arrowhead, which we can reconstruct from the transformed
vertices.

[ ],  [ ],  and [ ].
−1
1

1
−1

0
0

A = [ ].
−1   1   0
−1   −1      0

Let’s say we want to enlarge the arrowhead so that it is three times as tall
and two times as wide. This means we want to find a matrix T such that TA
represents the enlarged arrowhead.

Fig. 2.1.2 (generated from): Wolfram Alpha: Computational Knowledge Engine. (2009). Retrieved
March 26, 2016, from https://www.wolframalpha.com

Figure 2.1.2: Graph of the original arrowhead represented by the
matrix, A, and the enlarged arrowhead represented by the transformed

matrix, TA.

The matrix T represents a linear transformation that takes x to 2x and y to

3y, that is,

The matrix for this transformation is

What does T do to the vertices represented by the matrix A?

T[ ] = [ ].
x

y

2x

3y

T   =  [ ].
2  0
0 3

TA = [ ] [ ] = [ ].
2  0
0 3

−1  1 0
−1 −1   0

−2  2 0
−3  −3  0

Notice that the columns of this matrix are the vertices of the transformed
arrowhead in Figure 2.1.2.

Great! Now, lets try translating the original arrowhead up 5 units. Such a
simple transformation ought to be linear, too, right? Not quite.

If we attempt to find a matrix T such that

we find that this won’t work (try it!).

Since there is no matrix that works, and there is a one-to-one
correspondence between linear transformations and matrices, there can’t be
a linear transformation that performs the desired translation. At least, not as
we have set it up thus far.

Programmers would be pretty distraught if they had to give up linear algebra
to perform translations, which are everywhere in computer graphics
animations. Luckily, they don’t have to! the method of homogeneous
coordinates makes translations linear. Homogeneous coordinates are used
extensively in computer graphics for all linear transformations in order to be
able to include translations.

In homogeneous coordinates, each vector originally represented with two
coordinates is represented in three coordinates by appending

a 1 to it. So we’ll replace the original matrix A representing the arrowhead
by:

The linear transformation needed to

enlarge the arrowhead as we did before will now be represented by the
matrix:

Everything is the same as before except we have added the seemingly
superfluous extra row to A, and extra row and column to the linear
transformation matrix T.

T  [ ] =  [ ],
x

y

  x  
y + 5

A = .
⎡⎢⎣−1 1 0
−1 −1  0
1 1 1

⎤⎥⎦

T = ,  where now TA = .
⎡⎢⎣2  0  0
0 3 0
0 0 1

⎤⎥⎦ ⎡⎢⎣−2    2   0
−3 −3  0
1 1 1

⎤⎥⎦

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But now let’s examine the translation in which we move the arrowhead up 5
units. What we want is a linear transformation matrix T

that has the following effect:

In homogeneous coordinates, this can be represented by the matrix

so

our translation is now linear! The translated coordinates are given by the
transformed matrix:

Note that the last row of a transformation matrix T in homogeneous
coordinates will

always be a row of zeros with a 1 at the end, so that the matrix containing
the coordinates will always have its last row consisting entirely of 1’s.

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Let’s now use homogeneous coordinates to perform a trickier animation.
We’ll start with a matrix B representing the arrowhead starting with its tip at
the point (1,1) and we want to rotate it so that it is aimed at an object located
at position (–2, 4) as shown in Figure 2.1.3 below.

T = .
⎡⎢⎣x

y

1

⎤⎥⎦ ⎡⎢⎣ x

y + 5
1

⎤⎥⎦T = ,
⎡⎢⎣1   0  0
0  1   5
0 0 1

⎤⎥⎦TA = = .
⎡⎢⎣1   0  0
0  1 5
0  0 1

⎤⎥⎦⎡⎢⎣−1  1  0
−1  −1  0
1 1 1

⎤⎥⎦ ⎡⎢⎣−1  1  0
4 4 5
1 1 1

⎤⎥⎦

Fig. 2.1.3 (generated from): Wolfram Alpha: Computational Knowledge Engine. (2009). Retrieved
March 26, 2016, from https://www.wolframalpha.com

Figure 2.1.3: Rotating the arrowhead with its tip at (1,1) to aim it at an
object centered at (-2,4).

To do this, we will need to compose three linear transformations. This
technique is shown in the following video.

Video 2.1.1: Computing Linear Transformations using Homogeneous
Coordinates

Video by Dr. Lisa Korf: Computing Linear Transformations [© CCCOnline]

Video 2.1.1 Transcript: Computing Linear Transformations

As you may have concluded by now, homogeneous coordinates are an
indispensable tool in computer graphics. Click on the Check Your
Understanding tab to test yourself on what you’ve learned.

Introduction Computer Graphics Areas and Volumes Check Understanding
Discuss! References

0:00 / 12:34

https://media.ccconline.org/ccco/2016Master/MAT255/M2/video_2_1_transcript.txt

EXPLORATION OF MATRIX ALGEBRA AND

DETERMINANTS

Introduction Computer Graphics Areas and Volumes Check Understanding
Discuss! References

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Determinants of 2 by 2 and 3 by 3 matrices have a geometric interpretation
in terms of the areas and volumes of the two- or three-dimensional figures
being transformed. In particular, the determinant is a measure of areas and
volumes, as seen in the following theorem.

Theorem 2.2.1. Given a 2 by 2 matrix A whose column vectors represent
two adjacent sides of a parallelogram, the area of the
parallelogram.

Similarly, given a 3 by 3 matrix A whose column vectors represent three
edges of a parallelepiped emanating from the same vertex,
the volume of the parallelepiped.

The following video gives a short geometric proof for the 2 by 2 case. The 3
by 3 case can be proven similarly.

Video 2.2.1: Proof of Theorem 2.2.1

|det A| =

|det A| =

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https://app.readspeaker.com/cgi-bin/rsent?customerid=12160&url=https%3A%2F%2Fmycourses.cccs.edu%2Fcontent%2Fenforced%2F356437-S_CCA_MAT2540X00_202520%2FModule%252002%2FM2_Exploration.html&lang=en_us&readid=d2l_read_element_1

Video by Dr. Lisa Korf: Proof of Theorem 2.2.1 [© CCCOnline]

Video 2.2.1 Transcript: Proof of Theorem 2.2.1

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In the next video example, we’ll apply Theorem 2.2.1 to calculate the area of
the parallelogram with vertices given by the points (2, –2), (–3, 0), (1, 4)
and (–4, 6).

Video 2.2.2: Using the Determinant to Compute the Area of a
Parallelogram

0:00 / 4:43

https://media.ccconline.org/ccco/2016Master/MAT255/M2/proof_2_2_transcript.txt

Video by Dr. Lisa Korf: Using the Determinant to Computer the Area of a Parallelogram [©
CCCOnline]

Video 2.2.2 Transcript: Using the Determinant to Compute the Area of a
Parallelogram

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Determinants can also be used to calculate how the area or volume of a
figure changes when a linear transformation is applied to it.
Theorem 2.2.2. For a linear transformation T determined by a 2 by 2 matrix
A , if S is a figure in R3 with a finite area such as a disk, then:

For a linear transformation T determined by a 3 by 3 matrix A, if S is a
figure in R3 with a finite volume such as a ball, then:

0:00 / 2:08

Area of T (S) = |det  A|  ⋅Area of S.

Volume of T (S) = |det A|  ⋅ Volume of S.

https://media.ccconline.org/ccco/2016Master/MAT255/M2/video_2_2_transcript.txt

https://media.ccconline.org/ccco/2016Master/MAT255/M2/video_2_2_transcript.txt

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The next example illustrates how Theorem 2.2.2 can be used to compute the
area bounded by an ellipse.

Video 2.2.3: Computing the Area Bounded by an Ellipse

Video by Dr. Lisa Korf: Computing the Area Bounded by an Ellipse [© CCCOnline]

Video 2.2.3 Transcript: Computing the Area Bounded by an Ellipse

This same technique could be extended to compute the volume bounded by
an ellipsoid. Check your understanding next in the Check Understanding
tab.

Introduction Computer Graphics Areas and Volumes Check Understanding
Discuss! References

0:00 / 4:47

https://media.ccconline.org/ccco/2016Master/MAT255/M2/video_2_2_ellipse_transcript.txt