MA1 Report

8/14/2019 MA1 Report

1/23

Assignment 1 [Thomas Sampson, BSc 2 Games Software Development, 2Y]

Assignment 1

Module 5572: Mathematics for Modelling and Rendering

Contents

2D Image

1. Obtaining the vertices

2. Getting the vertices into excel

3. Translation transformations

4. Scaling transformations

5. Shear transformations6. Rotation about the origin

7. Rotation around an arbitrary point

8. Reflection in the axis

9. Reflection in an arbitrary line

3D Image


2. Parallel Projection

3. Perspective Projection4. Translation transformations

5. Scaling Transformations

6. Rotation around an axis

7. Rotation around an arbitrary axis

8. Reflection in the standard planes

9. Reflection in an arbitrary plane


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2D Image

The following work in this section involves the creation and manipulation of a 2D

image. I will use Microsoft Excel to display this 2D image, and any effects producedby applying transformation matrix to the points of the 2D image. I have taken

screenshots at regular intervals to demonstrate the effects of different techniques

(such as rotation, scaling etc) and have described how I achieved each

transformation.


The first step of my work was to obtain the vertices (2D co-ordinates) of a 2D image.

Last year, in a similar assignment I chose to use graph paper and a pencil to find the

points of a simple shape. However, this year I decided to digitise this stage by using3ds Max. I mapped a the following jpeg image Figure 1 onto the x-y plane, then used

the line tool to plot my co-ordinates

Figure 1

Note: - In this 2D example, all my pen points fall where z=0, hence there is no third dimension.

Screenshots

Here are some screenshots of me tracing the image of the car with the line tool.


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2. Getting the vertices into Excel

Now I had the vertices I needed for my drawing, I used 3ds max to export the model

into a "Wavefront Object" See Figure 2 with the extension ".obj". This file type works

particularly well as it saves all the vertices as plain text, from which it was easy to

copy and paste them into excel. I did however have to be careful of splitting out the

separate components of the car to achieve breaks in my graph.

Figure 2 - Example of a Wavefront Object file

The Result

Here you can see the vertices imported

into Excel. Each row shows a different

component of the vehicle (body, window

etc). Here I have expressed the vertices

as columns and have added z=1 to the

homogeneous co-ordinates (this is

necessary when implementing the

transformations)

Here you can see the vertices mapped

onto a simple line graph in Microsoft

Excel. I have coloured the different

components for aesthetic purposes.


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3. Translation Transformations

The first transformation I will perform on the co-ordinates is a simple transformation.

For each example I plot the original co-ordinates on the graph (in pale grey) and the

transformed co-ordinates onto the graph in red.

As my 2D image is quite large, for the purpose of this demonstration I am applying a

transformation of (30 ,60) using the following matrix.

Note: - I am using the Derive application to screenshot any matrices from this point onwards.All matrix multiplications are achieved in Excel using theMMULT(array1, array2)function.

The Result

Here you can see translation matrix in

excel and the parameters to the right (x

and y translation amounts) along with

some of the transformed vertices below.

As you can see, the

transformation was

successful and had

translated the car 30 units

to the right and 60 units

upwards,


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4. Scaling Transformations

The next transformation I will perform on the co-ordinates is a scaling transformation.

To demonstrate scaling I have used the following matrix 2 matrices.

a)

b)

The Result

Here are the results the 2 scaling matrices have upon my image.

a)

b)

This matrix will produce a stretch of 1.5 in the y direction.

This matrix will produce a stretch of 0.5 in both the x and y

direction.


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5. Shear Transformations

The next transformation I will perform on the co-ordinates is a shearing. Shearing

deforms the shape and can be applied along both the x and y-axis, therefore my

spreadsheet takes 2 parameters. I will use the following to matrices to apply shears

to my image along each axis.

a)

b)

The Result

Here are the results the 2 shearing matrices have upon my image.

a)

b)

This matrix will produce a shear of value 1 along the x-axis.

This matrix will produce a shear of value 0.2 along the y axis.


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6. Rotation about the origin

The first rotation transformation I will demonstrate is rotation around the origin. This

is the most simple of rotations and rotates vertices around the point (0,0). To achieve

this we use the following standard matrix.

Note: - In all cases of , represents the amount in degrees or radians you wish to rotate in

an anti-clockwise direction. I will use radians in all my rotations as this unit is better supported

by Excel.

For the purpose of this demonstration I will rotate my shape by 45o or /4cACW

(anti-clockwise), using the following matrix.

Note: - I used the RADIANS() function in excel to convert my degrees parameter to radians as

the COS() and SIN() function will only accept angles in this form

The Result

100

0cossin

0sincos

100

0/4)cos(/4)sin(

0/4)sin(/4)cos(


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7. Rotation about an arbitrary point

The next rotation is slightly more advanced as it will rotate my object around an

arbitrary point and is not limited to rotation only around the origin. There is no single

matrix to achieve this effect so I will use a composition of 3 matrices to achieve the

desired result. The first matrix translates the arbitrary point to the origin, the second

matrix applies the rotation, and the third matrix translates back to the arbitrary point.

Example 1

My first example will rotate the car 45o or /4c CW (clockwise) around the point of the

back tyre, approximately (255,45). To do this I will multiply the following 3 matrices.

As we are working with column vectors, the order of which the matrices get applied

to the object vertices is right to left (hence why these seem to be in the reverse

order to my previous description).

Example 1 Result

100

4510

25501

100

0/4)cos(/4)sin(

0/4)sin(/4)cos(

100

4510

25501


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7. Rotation about an arbitrary point continued.

Example 2

To demonstrate moving the arbitrary point my second example works in the same

way as example 1 but this time rotates the car 90o

or /2c

ACW (anti-clockwise)around the front tyre, approximately point (90,45), using the following matrices.

Example 2 Result

100

4510

9001

100

0/2)cos(/2)sin(

0/2)sin(/2)cos(

100

4510

9001


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8. Reflection in the axis

Vertices can be reflected in both the x and y-axis, and as with rotation, standard

matrices exist for these transformations. The following matrices produce reflections

in each axis.

The Result

Here are the results the two reflection matrices have upon my drawing

This matrix on the left produces a reflection in

the x-axis, and the matrix on the right

produces a reflection in the y axis.

Here is the

reflection in

the x-axis

Here is the

reflection in

the y axis


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9. Reflection in an arbitrary line

Reflection in an arbitrary line is more sophisticated than a rotation in the axis. As

there could be an infinite number of arbitrary lines, there is no standard formula for

achieving reflection these lines. Instead, the way reflection in a line is achieved, is to

bring that line down to meet an axis. Once this has been completed we can then

reflect in the axis and move the line back to its original position. For the purpose of

this example I will define a line by using two points, a and b (although lines could be

expressed in other forms such as Cartesian or parametric).

Example Line

a) (1, 3)

b) (4, 7)

Step 1

The first step is to bring the line down the origin using a translation. In this case we

can translate the line down by [-1,-3] so that it lies on the point (0, 0), using the

following matrix.

Step 2

Next we need to work out the angle which the line makes with the x-axis (we could

work with the y-axis but both methods work just as well). To find this angle we take

our transformed co-ordinates which are now (0, 0) and (3, 4), and use them to form a

triangle as follows.

Now we can calculate the angle using simple trigonometry, so

=Tan-1(4/3)

=53o or =0.927c

3

4


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Step 2 Continued

The angle calculated can now be used in the following matrix to rotate the line to

align it with the x-axis. Notice the angle is negative as we wish to achieve a CW

rotation.

Step 3

Now we have our line aligned with the x-axis we can do a simple x-axis reflection as

seen before using the following matrix..

Step 4 & 5

Once the reflection has taken place we can now rotate and translate back to the

original angle and position, using the opposite matrices used in steps 1 and 2.

All together now

Here is the final set of matrices to achieve a reflection in the arbitrary line given. As

we are using column vectors the matrices are applied to the points from right to left.

100

0-0.927)cos(-0.927)sin(

00.927)sin(0.927)cos(

100

00.927)cos(0.927)sin(

00.927)sin(0.927)cos(

To rotate back into position To move backinto position

100

00.927)cos(0.927)sin(

00.927)sin(0.927)cos(

100

0-0.927)cos(-0.927)sin(

00.927)sin(0.927)cos(


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The Result

Here is the result of a few reflections in arbitrary lines. Again these screenshots are

taken from graphs in Microsoft Excel.


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3D Image

The following work in this section involves the creation and manipulation of a 3D

image. Again I will use Microsoft Excel to display this 3D image, and any effects

produced by applying transformation matrix to the points of the 2D image.


For my 3D drawing I decided to obtain my using pen, paper and a calculator, rather

than a 3D software package. The reason for doing so is that the export / import

process is quite time consuming and would be even more so when using 3D vertices.

My drawing is of a windmill consisting of a base, top, fans and a door. I used

Microsoft Paint as evidence of my calculations. See Figure 1

Figure 1

The Result

Here you can see the vertices

imported into Excel.

Here you can see the vertices mapped onto a simple

line graph in Microsoft Excel shown in a 2 point

perspective projection. I have coloured the different

components for aesthetic purposes.


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2. Parallel Projection

The first method of projection I will demonstrate is parallel projection. Here are some

screenshots of parallel projections, along with the matrices used to achieve them.

Orthographic

Isometric

Orthographic projection allows us to see the

image from the plane z=0 using the matrix below.

Isometric projection as shown to the left is

achieved in three steps. We first rotate around

the vertical axis (y) by 45oand then rotate

around the horizontal axis by 35.264. This is

then followed by an orthographic projection in

the plane z=0 (see matrices at bottom of page).

Note: - Angles are shown in radians.


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3. Perspective Projection

The next method of projection I will demonstrate is perspective projection. I will

demonstrate one, two and three point perspective projections and show the matrices

used to achieve them.

One-point perspective

To achieve one-point perspective projections we first have to translate the points

before applying a one-point perspective projection matrix. In this example I will

translate my points by (10, 10, 0) before applying the projection matrix. The

projection matrix takes the parameter d which I will assign the value 50 for the sake

of this example. Here are the matrices which when multiplied, give a one-point

perspective projection. I have included a screenshot to show the outcome.

Here we can see that a one-point

perspective projection can look

distorted. A projection of this kind

produces only one vanishing point in

the z direction, with all other linesparallel with either the x or y axis.


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Two-point perspective

To achieve two-point perspective projections we have to both translate androtate

the points about the y-axis before applying the projection matrix. For the purpose of

this example I will translate my points by (30,-10,0) and then rotate 160o around the

y-axis before applying the projection matrix which will receive the parameter of dwith

a value 200. This will be achieved by multiplying the following three matrices (which

will be applied to my points from right to left).

Three-point perspective

Finally I will demonstrate a three-point perspective projection. To achieve this

projection we add another step to those used in previous perspective projections. As

before we translate the image and rotate around the y-axis, but this time we add an

extra rotation about the x-axis before applying the final projection matrix. For this

example I will translate by (10.10.20), rotate 160o around the y-axis, and 10o around

the x-axis before projection, using the following matrices (applied from right to left).

As you can see, a two-point

perspective projection is a lot more

convincing than a one-point

perspective projection, and provides

us with two vanishing points rather

than one. Here only the vertical lines

are parallel (with the y axis).


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Three-point perspective continued...

4. Translation transformations

Translation in 3D works very much the same was as in 2D but this time allows us to

translate in the z-axis in addition to the x and y-axis. Below is the standard 3D

translation matrix where a b and c are replaced with the amount you wish to translatein x y and z axis respectively.

Here I have applied the three point

perspective. It is difficult to see this

properly with my building as there are

not many vertical lines, but where

there are (such as on the roof, and

the side fans) you can see that they

all point to a vanishing point

somewhere beneath the image. This

projection of course provides us with

three vanishing points.

This is an example of a

translation of (20,10,10).

Note: - From here on I will be

using a two-point perspective

to show my 3D model being

translated, rotated and

reflected. This means that the

projection will be the final

step in each matrix

multiplication chain,

although I will not include

this in my examples.


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5. Scaling transformations

As with translations we can now add a third dimension to our scaling matrix. We use

the following standard matrix to achieve scaling in three dimensions. The letters a b

and c are replaced with the amount to scale in the x y and z direction respectively.

6. Rotation around the axis

As with 2D axis rotation there are a set of standard matrices for rotating 3D pointsaround the axis, with a third matrix allowing us to rotate around the z axis. The three

standard matrices are as follows and produce the following results...

This is an example of scaling

with a stretch of 1.5 in the xdirection, 0.5 in the y direction

and 2 in the z direction.

Rotation about the x-axis

In this example x = 45o

Rotation about the y-axis


Rotation about the z-axis



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7. Rotation around an arbitrary axis

Next I will demonstrate how to rotate around an arbitrary axis. This will allow me to specify any two

points, which will be joined to make an axis, then use the necessary transformations to rotate my

vertices around this axis in 3D space.

Step 1

The first step is to take the two points which define the line, and translate them such that one of the

points lies on the origin (0,0). If we define our points as (x1,y1,z1,x2,y2,z2) then we can use the

following matrix to achieve a shift to the origin.

Step 2

The way in which the rotation works it to make our arbitrary line align perfectly with one of the 3 axis

(for which we know standard rotation matrices) and carry out our rotation (followed by reversing the

alignment steps). In this step I need to find the angle the arbitrary line makes with the z axis. To do

this I take my line in component form ( x2i+y2j+z2k ) and the line of the z axis ( -k ) and find the dot

product of the two. This will give me a result . I can now rotate around the y axis to ensure my

arbitrary line lies on the y-z plane using the following matrix

Step 3

Now that my arbitrary line lies on the y-z plane a single rotation about the x axis will align my line

perfectly along the y axis. To find the angle I need to rotate around the x-axis I will take my new line

(after the previous matrix has been carried out) and the y axis ( j ) and find the dot product. This will

give me a result . I can now rotate about the x axis using the following matrix

Step 4

Step 4 is to actually carry out the rotation I wish to achieve ( around the y axis as my line is now

aligned perfectly ). To do this I use the following matrix


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Steps 5 6 & 7

Now that the rotation has taken place all that is left to do is revers the previous alignment process by

rotating and translating my arbitrary line back to where it started. This can be achieved by applying

the following three matrices (multiplied from right to left)

The Result

Here are some screenshots of rotations of 45o

around arbitrary lines.

8. Reflection in the standard planes

As with rotation around the axis, reflection in axis planes is produced using a standard set of

matrices, one for each plane. The three standard matrices are as follows and produce the

following results...

Reflection in the xy-plane Reflection in the xz-plane Reflection in the yz-plane


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9. Reflection in an arbitrary plane

The final transformation in my report is a reflection in an arbitrary plane. There are different

ways of defining a plane, but for my example I will specify my plane in component form,

specifying the distance of the plane from the origin (a+b+c), and specifying two lines which

lay on the plane.

Example plane

a+b+c + x1+y1+z1 + x2+y2+z2

Step 1

The system used to reflect in an arbitrary plane is quite similar to the system used to

rotate around an arbitrary line. Again, we begin by translating the plane to the origin.

This translation is easy to calculate as we simply reverse the effects of a b and c

using the following matrix.

Step 2

The second step is to find the normal to the plane. This allows us to determine which

way the plane is pointing/facing. To find the normal to any plane we take the cross

product of any two non-parallellines on that plane. In this case we can use the twolines specified by the plane (x1+y1+z1 and x2+y2+z2). This calculation will give us the

equation of the normal, in the form

nx+ny+nz

Step 3

Similarly to the rotation around an arbitrary line, we now need to align the normal

perfectly on an axis as doing so will allow us to reflect in a standard plane. Making

the y component of the normal 0 allows us to find the angle which the normalmakes with the z axis by finding the dot product. We can then rotate around the y

axis to bring the normal onto the y-z plane using the following matrix.


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