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### Cartoon Transformations

### Rotations

By: Justin Fernandez, Fiona McClean,

Sebastian Quiana, Eric Spiniello, and Wendy Star

Rotations – Fiona McClean Looney Tunes

- Reflections – Wendy Starr Simpsons
- Translations – Eric Spiniello Tom and Jerry
- Tessellations – Justin Fernandez Scooby Doo
- Dilations – Sebastian QuianaSpongebob

By: Fiona McClean

A rotation is an isometry where shapes rotate around a fixed point in a circular motion, whether clockwise or counter clockwise

- 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: the fixed point of a rotation
- P 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

(0,0)

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)

A

B

A’

D’

B’

C’

C

D

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)

The top of Tweety’s bird cage has rotational symmetry. It can be mapped onto itself at 36 degrees.

"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>.

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.

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

Line of SymmetryIn a non regular polygon, one must just count.

3 Sides

1 Line

4 Sides

4 Lines

Rx-axis (x,y) = (x, -y)

Ry-axis (x,y) = (-x,y)

Ry=x (x,y) = (y,x)

Ry=-x (x,y) = (-y,-x)

Pre-Image over line, find coordinatesTo 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

Equation Line of ReflectionTo 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

A (-1,5)

B (5,1)

A’ (-1,-5)

C (4,0)

Minimum DistanceB

C

What is the equation of the line A’B?

A’

If a character from the Simpsons were to look into a mirror they would see their face reflected back at them.

Real Life ApplicationGo 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).

http://t3.gstatic.com/images?q=tbn:ANd9GcTfjjBz37-c-6b2x_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_xavY8https://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

BibliographyTranslation: 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.

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 ExamplesYou 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)

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).

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.

What is the rule for the translation you just made?

- How would you write that rule in component form?

In cartoons, translations are everywhere. For example, as Tom runs after Jerry, both characters are translating and moving across the screen.

Real World Applicationstomjerrynew.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)

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

What's a TesserrationRaggy?- Around any vertex or corner point in a tessellation the measure of all angles must equal 360 degrees

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

Jeepers Gang, look at that Tessellation!- 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

Find a picture that you want to tessellate like this one. A square, rectangle, right triangle, or regular triangle would be the easiest.

- Start by placing it into a new Photoshop document
- 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

How to Create a Scooby Snack (Tessellation)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.

Dilation – Transformation that produces a shape that is different in size

- Scale Factor – Ratio of corresponding sides of an image over a pre-image (K)
- Reduction – If the scale factor is less than one
- Enlargement – If the scale factor is greater than one

Center of Dilation – A fixed point where all points are dilated

- Equation: Dk(x,y)=(kx,ky)
- K = OP’/OP

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

1/2

A

Multiplying with Matrices

B

c

A B C

X = 5 7 3

Y = 6 3 3

1/2

Example of a ReductionA’ B’ C’

X = 2.5 3.5 1.5

Y = 3 1.5 1.5

Scale Factor

2

Multiplying with Matrices

A

A B C

X = 5 7 3

Y = 6 3 3

B

c

2

Example of a EnlargementA’ B’ C’

X = 10 14 6

Y = 12 6 6

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

Real Life Application- 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

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

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?

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

- 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’

- 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

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