Cartoon
Transformations
By: Justin Fernandez, Fiona McClean,
Sebastian Quiana, Eric Spiniello, and Wendy Star
Table of Contents
•
•
•
•
•
Rotations – Fiona McClean Looney Tunes
Reflections – Wendy Starr Simpsons
Translations – Eric Spiniello Tom and Jerry
Tessellations – Justin Fernandez Scooby Doo
Dilations – Sebastian Quiana Spongebob
Rotations
By: Fiona McClean
Rotations
A rotation is an isometry where shapes rotate around a fixed point
in a circular motion, whether clockwise or counter clockwise
Rotations Vocabulary
Angle of Rotation: rays drawn from the center of rotation to a point
and its image form an angle
-If center of rotation is origin:
•
•
•
•
R90° (x,y) = (-y,x)
R180° (x,y) = (-x,-y)
R270° (x,y) = (y,-x)
R-90° (x,y) = (y,-x)
Center of Rotation:
P
the fixed point of a rotation
Point P is the center of rotation
Rotational Symmetry: when a figure can be mapped onto itself by a
clockwise rotation of 180 degrees or less
- An equilateral triangle can be mapped onto itself by 120 degrees
Center of Rotation:
(0,0)
A
B
D
C
D
’
C’
A’
B’
Angle of Rotation:
270°
Vertices:
A (-4,4,)
A’ (4,4)
B (-2,4)
B’ (4,2)
C (-2,2)
C’ (2,2)
D (-4,2)
D’ (2,4)
Center of Rotation:
Point F
B
A
C
E
F
D
Angle of Rotation:
80 degrees
Help Tweety
Tweety wants to go into
his cage. Rotate Tweety
110 degrees about point P,
(9,4), so that he is in his
cage. Find A’, B’, C’, D’,
and E’.
Vertices:
A (16.5, 7)
B (14, 6.5)
C (12.5, 7)
D (13, 9)
E (14.5,
9.5)
Real Life Application
The top of Tweety’s bird cage has rotational symmetry. It can be
mapped onto itself at 36 degrees.
Bibliography
• "Looney Tunes." SAT 400. N.p., n.d. Web. 23 Apr. 2013.
<http://www.sat400.com/
satlooney.html>.
• "EK Success Wavy Circle Large Punch." BGPayne Crafts. N.p., n.d.
Web. 23 Apr.
2013. <http://www.bgpaynecrafts.co.uk/products/
21307-ek-success-wavy-circle-large-punch.aspx>.
Reflections
By: Wendy Star
Vocabulary
• Reflection – an image over a line, that almost acts like a
mirror.
• Line of Reflection – the which acts like a mirror in a
reflection.
• Line of symmetry – a figure that can be mapped onto
itself by a reflection in the line.
• Isometry – transformation which the two figures are
congruent.
Line of Symmetry
In a regular polygon, the
number of lines of symmetry
is equal to the number of
side.
3 Sides
3 Lines
4 Sides
2 Lines
In a non regular
polygon, one must
just count.
4 Sides
4 Lines
3 Sides
1 Line
All of their faces have one line of symmetry.
Pre-Image over line, find coordinates
Equations need:
Rx-axis (x,y) = (x, -y)
Ry-axis (x,y) = (-x,y)
Ry=x (x,y) = (y,x)
Ry=-x (x,y) = (-y,-x)
Equation Line of Reflection
To find the line of reflection, you find the midpoints, from
matching vertexes, and graph the line. That will be the line of
reflection.
Line of
Reflection
Marge Simpson Reflected
Minimum Distance
To find the minimum distance, you reflect one of the initial
points (point A), then you connect A’ to point B. where that
line crosses the x-axis will be the minimum distance point
C.
A
B
A (-1,5)
B (5,1)
A’ (-1,-5)
C (4,0)
C
A’
What is the
equation of the
line A’B?
Real Life Application
If a character from the Simpsons were to look into a mirror they
would see their face reflected back at them.
GSP Activity
•
•
•
•
Go onto GSP and make sure you have graph up.
Then plot A(-1,-2) and B (8,-4).
Next find the minimum distance and the equation of A’B.
Do the same for A (1,4), B (8,3).
Bibliography
All Slides:
http://t3.gstatic.com/images?q=tbn:ANd9GcTfjjBz37-c6b2x_EImq34uX60zmXCwN7Pyf7x91AFdhW727Ju:upload.wikimedia.org/wikipedia/en/3/33/All_Simpsons_characters.jpg
Slide 1: http://images1.wikia.nocookie.net/__cb20100602025911/simpsons/images/6/65/Bart_Simpson.png
Slide 4:
https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcS4qxV1o2Dk4yIHd8rC5t_oMcpdrVdGk4491jfC8FDNlLKzULw9
https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcQljPs8VRIbBCHfCibsqAxm3Qw0NaglTlxHWqMimdZD1z_xavY8
https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcTZn-JGnr9jvpfHqPTjslYUKPcxD3vg6eGFCtiULC5Fl7gnWkS7xw
Slide 6:
http://www.regentsprep.org/Regents/math/geometry/GT1/xgraph.gif
http://www.regentsprep.org/Regents/math/geometry/GT1/PtGraph.gif
Slide 7:
https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcR9704cU93y6UevI-_uuXKUnv52ywQQh2ZkxPiH0Av4oOjUgbRu
https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcSn4iIp-Z4GW5FdkE62UNWhOnne5fIs1kEEWn2YzUw_bxuHqXGe
Slide 9:
http://slacktory.com/wp-content/uploads/2011/10/Marge-vs-Girl-at-Mirror.jpg
https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRtifkK3QsjzDWP92u1I3jtRYbWG1tRU_i6Yf_ph7b3agfiaHw6
Slide 10:
Chapter 7 Resource Book Lesson 7.2
Tom and Jerry’s
Translations
By: Eric Spiniello
Words to Know:
• Translation: a type of transformation where every point of a pre image is
moved a certain distance is a certain direction to form an image. The image
is congruent to the pre image, it is just moved.
• Initial Point: The starting point of a vector.
• Terminal Point: The end point of a vector
• Vector: a quantity that has both magnitude and direction.
• Component Form: is made up by the horizontal and vertical components of
a vector. For example, the rule (x,y)>(x+2,y-3) becomes <(2,-3)> in
component form.
• Coordinate notation: tells you the distance on the x and y axis you should
move each point. For example, (x,y)>>>(x+a,y+b). “A” represents the
amount of units you moved on the x axis and “B” represents the amount that
you moved on the y axis.
Mammy Two Shoes Mathematical Examples
• In this example, each point slides 7 units left and 3 down.
This means that the rule is (x,y)>>(x-7,y-3). In
component/vector form it would be <-7,-3> and in
coordinate notation (x,y)>>(x-7,y-3).
Mammy Two Shoes Mathematical Examples
• You can also find this by using matrices. First, you must take the
coordinates of A B C and D and record them in a matrix. The x
coordinate plots go on the top, with the y on the bottom.
[A B C D]
[A B C D]
[A B C D]
[2 4 5 2] + [-7 -7 -7 -7] = [-5 -3 -2 -5]
[4 4 2 1] + [-3 -3 -3 -3] = [ 1 1 -1 -2]
• Since the rule is (x,y)>>(x-7,y-3), we added -7 to the x coordinates and
-3 to the y coordinates. This tells us that the new coordinates for the
image are A’= (-5,1) B’= (-3,1) C’= (-2,-1) D’= (-5,-2)
Tom’s Translation Concepts
• If you are given a pre image at
point (3,-2) and a rule
(x,y)>>(x+5,y-2) then you would
start at point (3,-2) and count 5
units to the right and 2 units down
on a coordinate plane. So the
coordinates of the image would be
(8,-4).
• If you are given the image at point
(8,-4) and a rule (x,y)>>(x+5,y-2),
then you would subtract 5 from 8
and add 2 to -4. This would make
the coordinates of the pre image
(3,-2).
Jerry’s GSP Activity
• Under graph, click “show grid”.
• Create any shape of your
choice. Label the points.
• Highlight your shape.
• Go to the “Transform” window
and select “Translate”.
• On the new window select
“Rectangular” under
“Translation Vector”.
• Write in 7cm for the horizontal
and 2cm for the vertical fixed
distances. This will be the
number of units your new
image will translate.
• Click “Translate”.
• Your new image is a translation
from the original pre image. It
should look like the example.
Jerry’s GSP Activity
Question
1. What is the rule for the translation you just made?
2. How would you write that rule in component form?
Real World Applications
• In cartoons, translations are everywhere. For example, as
Tom runs after Jerry, both characters are translating and
moving across the screen.
Bibliography
• tomjerrynew.blogspot.com (Title Photo)
• http://atminhd.com/tom-and-jerry-wallpaper-download-hd.html (Tom
and Jerry Second Slide)
• http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm
(Translation Diagram)
• http://en.wikipedia.org/wiki/Mammy_Two_Shoes (Mammy Two Shoes)
• protagonist.wikia.com (Tom)
• mugen.wikia.com (Jerry)
• http://www.goldenagecartoons.com/reviews/2008/tjtales4/ (Tom and
Jerry Confused)
• www.toptimelinecovers.com (Tom chasing Jerry)
• http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm
(Math Information)
• Geometry Textbook (Math Information)
Tessellations
By: Justin Fernandez
What's a Tesserration Raggy?
• A tessellation is the process of creating a two-dimensional plane using
the repetition of a geometric shape with no overlaps and no gaps
Example:
In this case four
90 degree angles
• Around any vertex or corner point in
a tessellation the measure of all
angles must equal 360 degrees
Jeepers Gang, look at that Tessellation!
• There are many tessellations that appear everywhere
• The most common being floor tilling The tessellation made in this
floor tilling is made of regular
squares alternating from white
to black
• This is made of rectangular bricks on a
wall on a street or of a house
• This tessellation is know as a 4.4.4
tessellation because of the amount of
shapes and their # of sides
4
4
4
Scooby Doodles
Rings Ry Ridn’t Rake
(things I didn’t make)
How to Create a Scooby
Snack (Tessellation)
1.
Find a picture that you want to tessellate like
this one. A square, rectangle, right triangle, or
regular triangle would be the easiest.
2. Start by placing it into a new Photoshop
document
1. The size should be U.S.
paper size which is selected when
you select File New then in the
Present drop down menu select U.S.
Paper
3. To get the picture in the document copy the picture form any online
source and then in the Edit drop down menu on the top left select paste
and the picture should be right there.
4. After this press the Crtl key simultaneously with the T key and the
picture should be able to be resized. Hold shift and on the corners
adjust it to your proffered size. After this press Crtl and the D key
5. Move the picture with the tool by pressing v and move it to the top
left leaving room in between the top and the left edge of the paper
6. Then ¾ down the right hand side there is a tab called layers select
that
7. Right click the layer that says Layer 1 and select duplicate
8. The new image should appear right on top of the other one, just take
this image and move it to be right next to the other one to the right
9. Keep doing this until you have enough to make a row across the
paper but not touching the right edge
10. Then select on the layers tab the top most image, right click and
select merge down until all you have is the one layer and the
background
11. Then duplicate this layer comprised of all the copied pictures and
duplicate that. Then place this copy underneath the other image
12. Keep doing this until the whole page is filled. The final project
should look somewhat like this.
Dilations
By: Sebastian Quiana
Terms To Know
• Dilation – Transformation that produces a shape that is different in size
• Scale Factor – Ratio of corresponding sides of an image over a preimage (K)
• Reduction – If the scale factor is less than one
• Enlargement – If the scale factor is greater than one
Terms to know Cont.
• Center of Dilation – A fixed point where all points are dilated
• Equation: Dk(x,y)=(kx,ky)
• K = OP’/OP
Properties preserved
•
•
•
•
•
Angle measures remain the same
(Parallelism) Parallel lines remain parallel
(Colinearity) Points stay on the same lines
(Midpoint) Midpoints remain the same in each figure
(Orientation) Lettering order remains the same
Example of a Reduction
Scale Factor
1/2
Multiplying with Matrices
A
X=
Y=
B
C
5
6
A’
X=
Y=
7
3
B’
2.5
3
3
3
C’
3.5
1.5
1.5
1.5
1/2
Example of a Enlargement
Enlargement
Scale Factor
2
Multiplying with Matrices
A
B
C
X= 5
7
3
Y= 6
3
3
A’
B’
C’
X = 10
14
6
Y = 12
6
6
2
Real Life Application
It has been discovered that a full grown Box
Jellyfish can be 300 cm long and 25 cm wide. It has also
been found that a baby Box Jellyfish can measure 15 cm
long.
15 cm
300 cm
• Find the scale Factor of the length
• Using the Scale factor find the width
of the baby jellyfish
• Is this a reduction or an enlargement
GSP Activity
•
•
•
•
•
•
•
•
•
First Click ‘graph’ and then click on ‘form grid’
Next Create a triangle
Label the triangle ABC
Measure the lengths of the triangle
Highlight the triangle and click ‘transform’ then ‘dilate’
Use a scale factor of ½
Make this triangle A’B’C’
Measure the Lengths of the new triangle
Move the triangle around examining what happens
GSP Questions
• Is the new triangle a reduction or a enlargement?
• What happens when you move triangle ABC?
• If you dilate triangle A’B’C’ with the same scale factor
what happens?
• What Happens when you move the original triangle at the
end of the steps above?
Bibliography
•
•
•
•
•
•
•
•
•
http://thewirecutter.com/reviews/best-tv-panasonic-st60/
spongebob.wikia.com
poohadventures.wikia.com
cartoons.wikia.com
mycrappyneighbor.com
en.wikifur.com
spongebob.neoseeker.com
www.mommypeach.com
spongebob.wikia.com
Answers
•
•
•
•
•
•
•
•
• Dilation Answers
Real Life applications answer – 300/15 = 20/1
1.25 cm
Enlargement
GSP Answers
When you move triangle ABC triangle A’B’C’ should move making
the side lengths ½ triangle ABC
Reduction
The new triangle will have half the side lengths of triangle A’B’C’
A’B’C’ should be half the side length of Triangle ABC and The new
triangle should be half the side lengths triangle A’B’C’
Answers
• Translation Answers
• 1: (x,y)>>>>(x+7,y+2)
• 2: <7,2>
• Reflection Answers
• What is the equation of the line?
y=x-4
• A(-1,-2) and B (8,-4), C (2,0). y= -2/9x+4/9
• A (1,4), B (8,3), C (5,0). y= -1/7x+5/7
Answers
• Rotation Answers
•
•
•
•
•
•
Vertices:
A’ (4,10)
B’ (5,8)
C’ (5,6)
D’ (3,6)
E’ (3,8)