+
+
+Complexity
+
+
Ask the student to justify the complexity of the functions. It must be at
+most O(n) in time and O(n) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Add
+
+
Check the behaviour of the vector addition with the following parameters:
+
+
+- '[0, 0]' and '[0, 0]' give '[0, 0]'
+- '[1, 0]' and '[0, 1]' give '[1, 1]'
+- '[1, 1]' and '[1, 1]' give '[2, 2]'
+- '[21, 21]' and '[21, 21]' give '[42, 42]'
+- '[-21, 21]' and '[21, -21]' give '[0, 0]'
+- '[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]' and '[9, 8, 7, 6, 5, 4, 3, 2, 1, 0]' give '[9, 9, 9, 9, 9, 9, 9, 9, 9, 9]'
+
+
+Check the behaviour of the matrix addition with the following parameters:
+
+
+- '[[0, 0], [0, 0]]' and '[[0, 0], [0, 0]]' give '[[0, 0], [0, 0]]'
+- '[[1, 0], [0, 1]]' and '[[0, 0], [0, 0]]' give '[[1, 0], [0, 1]]'
+- '[[1, 1], [1, 1]]' and '[[1, 1], [1, 1]]' give '[[2, 2], [2, 2]]'
+- '[[21, 21], [21, 21]]' and '[[21, 21], [21, 21]]' give '[[42, 42], [42, 42]]'
+
+
+Feel free to perform more tests on your own
+
+
+
+Subtract
+
+
Check the behaviour of vector subtraction with the following parameters:
+
+
+- '[0, 0]' and '[0, 0]' give '[0, 0]'
+- '[1, 0]' and '[0, 1]' give '[1, -1]'
+- '[1, 1]' and '[1, 1]' give '[0, 0]'
+- '[21, 21]' and '[21, 21]' give '[0, 0]'
+- '[-21, 21]' and '[21, -21]' give '[-42, 42]'
+- '[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]' and '[9, 8, 7, 6, 5, 4, 3, 2, 1, 0]' give '[-9, -7, -5, -3, -1, 1, 3, 5, 7, 9]'
+
+
+Check the behaviour of matrix subtraction with the following parameters:
+
+
+- '[[0, 0], [0, 0]]' and '[[0, 0], [0, 0]]' give '[[0, 0], [0, 0]]'
+- '[[1, 0], [0, 1]]' and '[[0, 0], [0, 0]]' give '[[1, 0], [0, 1]]'
+- '[[1, 1], [1, 1]]' and '[[1, 1], [1, 1]]' give '[[0, 0], [0, 0]]'
+- '[[21, 21], [21, 21]]' and '[[21, 21], [21, 21]]' give '[[0, 0], [0, 0]]'
+
+
+Feel free to perform more tests on your own
+
+
+
+Multiply
+
+
Check the behaviour of vector scaling with the following parameters:
+
+
+- '[0, 0]' and '1' give '[0, 0]'
+- '[1, 0]' and '1' give '[1, 0]'
+- '[1, 1]' and '2' give '[2, 2]'
+- '[21, 21]' and '2' give '[42, 42]'
+- '[42, 42]' and '0.5' give '[21, 21]'
+
+
+Check the behaviour of matrix scaling with the following parameters:
+
+
+- '[[0, 0], [0, 0]]' and '0' give '[[0, 0], [0, 0]]'
+- '[[1, 0], [0, 1]]' and '1' give '[[1, 0], [0, 1]]'
+- '[[1, 2], [3, 4]]' and '2' give '[[2, 4], [6, 8]]'
+- '[[21, 21], [21, 21]]' and '0.5' give '[[10.5, 10.5], [10.5, 10.5]]'
+
+
+Feel free to perform more tests on your own
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n) in time and O(n) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Test the behaviour of linear combinations of vectors with the following parameters:
+
+
+- 'linear_combination([Vector::from([-42., 42.])], [-1.])' gives '[42., -42.]'
+- 'linear_combination([Vector::from([-42.]), Vector::from([-42.]), Vector::from([-42.])], [-1., 1., 0.])' gives '[0.]'
+- 'linear_combination([Vector::from([-42., 42.]), Vector::from([1.,
+3.]), Vector::from([10., 20.])], [1., -10., -1.])' gives '[-62., -8.]'
+- 'linear_combination([Vector::from([-42., 100., -69.5]), Vector::from([1., 3., 5.])], [1., -10.])' gives '[-52., 70., -119.5]'
+
+
+Feel free to perform more tests on your own.
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n) in time and O(n) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameters:
+
+
+- 'lerp(0., 1., 0.)' gives '0.'
+- 'lerp(0., 1., 1.)' gives '1.'
+- 'lerp(0., 42., 0.5)' gives '21.'
+- 'lerp(-42., 42., 0.5)' gives '0.'
+- 'lerp(Vector::from([-42., 42.]), Vector::from([42., -42.]), 0.5)' gives '[0.0] [0.0]'
+
+
+Feel free to perform more tests on your own.
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n) in time and O(n) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameters:
+
+
+- '[0, 0]' and '[0, 0]' gives '0'
+- '[1, 0]' and '[0, 0]' gives '0'
+- '[1, 0]' and '[1, 0]' gives '1'
+- '[1, 0]' and '[0, 1]' gives '0'
+- '[1, 1]' and '[1, 1]' gives '2'
+- '[4, 2]' and '[2, 1]' gives '10'
+
+
+Feel free to perform more tests on your own.
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the functions. It must be at
+most O(n) in time and O(n) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Euclidean norm
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[0]' returns '0'.
+- '[1]' returns '1'.
+- '[0, 0]' returns '0'.
+- '[1, 0]' returns '1'.
+- '[2, 1]' returns '2.236067977'.
+- '[4, 2]' returns '4.472135955'.
+- '[-4, -2]' returns '4.472135955'.
+
+
+Feel free to perform more tests on your own.
+
+
+
+Manhattan norm
+
+
+- '[0]' returns '0'.
+- '[1]' returns '1'.
+- '[0, 0]' returns '0'.
+- '[1, 0]' returns '1'.
+- '[2, 1]' returns '3'.
+- '[4, 2]' returns '6'.
+- '[-4, -2]' returns '6'.
+
+
+Feel free to perform more tests on your own.
+
+
+
+Supremum norm
+
+
Test the function with several different vectors. Each time, the function
+must return the component of the vector with the greatest value.
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n) in time and O(n) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameters:
+
+
+- '[1 0]' and '[0 1]' gives '0'
+- '[8 7]' and '[3 2]' gives '0.9914542955425437'
+- '[1 1]' and '[1 1]' gives '1'
+- '[4 2]' and '[1 1]' gives '0.9486832980505138'
+- '[-7 3]' and '[6 4]' gives '-0.5462677805469223'
+
+
+Since the order of the parameters doesn't matter (the function is said to be
+commutative), the function must return the same result if you swap them.
+
+Feel free to perform more tests on your own.
+
+
+
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameters:
+
+
+- '[0 0 0]' and '[0 0 0]' gives '[0 0 0]'
+- '[1 0 0]' and '[0 0 0]' gives '[0 0 0]'
+- '[1 0 0]' and '[0 1 0]' gives '[0 0 1]'
+- '[8 7 -4]' and '[3 2 1]' gives '[15 -20 -5]'
+- '[1 1 1]' and '[0 0 0]' gives '[0 0 0]'
+- '[1 1 1]' and '[1 1 1]' gives '[0 0 0]'
+
+
+Feel free to perform more tests on your own. When giving two vectors to
+the function, imagine them creating a plane. Then, the function must return
+a vector that is orthogonal (perpendicular) to that plane.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n^3) in time and O(n^2) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[0, 0], [0, 0]]' and any vector of dimension two. The function must always return vectors with only zeros in it.
+- '[[1, 0], [0, 1]]' and any vector of dimension two. The function must always return the same vector as given in parameter.
+- '[[1, 1], [1, 1]]' and '[4, 2]'. The function must return '[6, 6]'.
+- '[[2, 0], [0, 2]]' and '[2, 1]'. The function must return '[4, 2]'.
+- '[[0.5, 0], [0, 0.5]]' and '[4, 2]'. The function must return '[2, 1]'.
+
+
+Feel free to perform more tests on your own
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at most O(n) in time.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[0, 0], [0, 0]]' returns '0'
+- '[[1, 0], [0, 1]]' returns '2'
+- '[[1, 2], [3, 4]]' returns '5'
+- '[[8, -7], [4, 2]]' returns '10'
+- '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '3'
+
+
+Feel free to perform more tests on your own
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n^2) (value assignments) in time and O(n^2) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[0, 0], [0, 0]]' returns '[[0, 0], [0, 0]]'
+- '[[1, 0], [0, 1]]' returns '[[1, 0], [0, 1]]'
+- '[[1, 2], [3, 4]]' returns '[[1, 3], [2, 4]]'
+- '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]'
+- '[[1, 2], [3, 4], [5, 6]]' returns '[[1, 3, 5], [2, 4, 6]]'
+
+
+Feel free to perform more tests on your own
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n^3) in time and O(n^2) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[0, 0], [0, 0]]' gives '[[0, 0], [0, 0]]'
+- '[[1, 0], [0, 1]]' gives '[[1, 0], [0, 1]]'
+- '[[4, 2], [2, 1]]' gives '[[1, 0.5], [0, 0]]'
+- '[[-7, 2], [4, 8]]' gives '[[1, 0], [0, 1]]'
+- '[[1, 2], [4, 8]]' gives '[[1, 2], [0, 0]]'
+
+
+Feel free to perform more tests on your own
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n^3) in time.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[0, 0], [0, 0]]' returns '0'
+- '[[1, 0], [0, 1]]' returns '1'
+- '[[2, 0], [0, 2]]' returns '4'
+- '[[1, 1], [1, 1]]' returns '0'
+- '[[0, 1], [1, 0]]' returns '-1'
+- '[[1, 2], [3, 4]]' returns '-2'
+- '[[-7, 5], [4, 6]]' returns '-62'
+- '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '1'
+
+
+Feel free to perform more tests on your own
+
+
+
+Explanations
+
+
Ask the student to explain:
+
+
+- What happens when the determinant of a matrix is '0'.
+- What the determinant represents geometrically in the vector space
+(ie, what happens after using the matrix for a linear transformation,
+and what does the determinant describe)
+
+
+If they cannot explain it, the evaluation ends here.
+
+
+
+
+
+
+Complexity
+
+
Ask the student to justify the complexity of the function. It must be at
+most O(n^3) in time and O(n^2) in space.
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[1, 0], [0, 1]]' returns '[[1, 0], [0, 1]]'
+- '[[2, 0], [0, 2]]' returns '[[0.5, 0], [0, 0.5]]'
+- '[[0.5, 0], [0, 0.5]]' returns '[[2, 0], [0, 2]]'
+- '[[0, 1], [1, 0]]' returns '[[0, 1], [1, 0]]'
+- '[[1, 2], [3, 4]]' returns '[[-2, 1], [1.5, -0.5]]'
+- '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]'
+
+
+Feel free to perform more tests on your own. To check the result, you can
+multiply it by the matrix you gave as parameter and it must give (approximately)
+the identity matrix (However, avoid testing matrices that are not invertible).
+
+
+
+
+
+
+Basic tests
+
+
Check the behaviour of the function with the following parameter:
+
+
+- '[[0, 0], [0, 0]]' returns '0'
+- '[[1, 0], [0, 1]]' returns '2'
+- '[[2, 0], [0, 2]]' returns '2'
+- '[[1, 1], [1, 1]]' returns '1'
+- '[[0, 1], [1, 0]]' returns '2'
+- '[[1, 2], [3, 4]]' returns '2'
+- '[[-7, 5], [4, 6]]' returns '2'
+- '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '3'
+
+
+Feel free to perform more tests on your own
+
+Check the use of forbidden mathematical functions (see the subject).
+
+
+
+Explanations
+
+
Ask the student to explain what the rank of a matrix represents.
+
+If they cannot explain it, the evaluation ends here. You can use the
+internet to check the answers.
+
+
+
+
+
+
+Projection
+
+
Build several matrices with several FoVs (convert the value in radians before
+passing it to the function):
+
+
+- 100 degrees
+- 70 degrees
+- 40 degrees
+
+
+Then, test the matrices in the projection utility given in the attachements.
+
+Also, try testing with several different combinations of near/far values (near
+must stay smaller than far) and different ratios (the default is 1).
+
+A lower FoV must reduce the angle of view.
+
+Changing the ratio must distort the image.
+
+Different values of near and far must change the distance from the camera at
+which objects disappear from the screen.
+
+Ask the student to explain what each component of the matrix represents.
+
+
+
+
+
+
+Lots of tests
+
+
For this exercise, the student must have recoded all the previous functions (except for ex14),
+or used the generic structure of the code, to provide the use of complex
+numbers as scalars. The student should be able to explain how the operations
+of complex numbers work (geometrically).
+
+Reminder of the rules for complex numbers:
+
+
+- 'i^2 = -1'
+- '(a + bi) + (c + di) = (a + c) + (b + d)i'
+- '(a + bi) - (c + di) = (a - c) + (b - d)i'
+- '(a + bi) * (c + di) = (ac - bd) + (bc + ad)i'
+- '(a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)'
+
+
+Test every function, but with complex numbers, and check that they behave
+correctly. The student that has done this bonus should probably provide
+tests for complex numbers in his executables, and show them along with
+the correction for the regular exercises, if they wish to gain time.
+
+
+
+
+