Problem 1 (Matlab)

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This is the function file that you must have in order to proceed.

function DrawHouse( m )

%Define points and draw the house

house = [ -0.5 0.5 0.5 0.0 -0.5 -0.5; -0.5 -0.5 0.5 0.75 0.5 -0.5];

window = [ 0.1 0.3 0.3 0.1 0.1; -0.1 -0.1 0.3 0.3 -0.1 ];

door = [ -0.2 -0.2 -0.4 -0.4 ; -0.5 -0.2 -0.2 -0.5 ];

house(3,:) = ones(1, size(house, 2));

window(3,:) = ones(1, size(window, 2));

door(3,:) = ones(1, size(door, 2));

house = m * house;

window = m * window;

door = m * door;

plot(house(1,:), house(2,:), '-b' )

hold on;

plot(window(1,:), window(2,:), '-r' )

plot(door(1,:), door(2,:), '-k' )

end

Use a matrix to move a house around. Translate by (5, 1.5) and rotate by 25 degrees then try both combinations of rotating then translating

Deliverables:

Script to make the four plots

Make a translation matrix and check that it correctly translates

Make a rotation matrix and check that it correctly rotates

Also verify that it is orthogonal

Plot with 4 versions of the house: Translated, rotated, translated then rotated, rotated then translated

You must use your matrices to do this

Step by Step Instructions:

First make four plots and plot the house at the origin in each of them

Call DrawHouse with the identity matrix

Use axis equal to make it the right aspect ratio (not squished)

Put a title on the subplot but you can skip labels

Make a translation matrix

Use eye to make an identity matrix mTrans. Must be a 3x3 matrix

Set dx to be 5, dy to be 1.5

The upper right two elements of mTrans

(refer to lecture notes)

To check that you have the correct matrix, try

mTrans * [0;0;1]

You should get the column vector 5, 1.5, 1

See self check below

Use DrawHouse with mTrans to make the first picture

Make a rotation matrix

Use eye (again) to make an identity matrix mRot. Must also be a 3x3 matrix

Declare a variable for theta

Fill in the matrix; refer to lecture notes.

cos, sin

-sin, cos

Check that you have the correct matrixdot( mRot(1,:), mRot(2,:) ) is zero

as is every other dot product if the rows are different

dot( mRot(1,:), mRot(1,:)) is one

as is every other dot product if the rows are the same

mRot * mRot is the identity matrix (or close enough)

Math fact: This is the matrix multiplied by its transpose. One property of rotation matrices is that their transpose is their inverse i.e., rotating in the opposite direction. Youll notice that mRot is the same as mRot, except the minus sign on the sin is swapped

Note: you only need to check the conditions above for this labs hand-in, but in general, if youre making a rotation matrix you should ALWAYS check all of these.

Use DrawHouse with mRot to make the second picture

Now make the 3^rd and 4^th picturesNote: The matrix that is on the right is the one that happens first. So

mTrans * mRot

rotates then translates (yes, that seems backwards for those of us who read left to right)

Math fact II: This is a visual demonstration of the fact that matrix multiplication is not communitive A * B is not the same as B * A.