Study Guide 1: Mathematical foundations

Quizes are closed book. 30 minutes in lab.

Topics:

  • Vectors and points

  • Matrices, inverse matrix, matrix multiplication, identity matrix

  • Transformations, Euler angles, homogeneous coordinates

  • Trigonometry: sine, cosine, tangent

  • Dot and cross products

  • Line equations

  • Barycentric coordinates

Vectors and points

1) Consider the points \(a=(-4, -1, 0)\), \(b=(2, 2, 0)\), and \(c=(-1, 6, 0)\). Draw these points on a coordinate plane.

  • Compute the vectors \(v_{ab} = b-a\) and \(v_{ac} = c-a\). In the space above, draw them.

  • Compute and draw the sum of \(v_{ab} + v_{ac}\).

  • Compute and draw the vector \(-v_{ab}\). What the geometric interpretation of negating a vector?

  • Compute and draw the vector \(2v_{ac}\). What the geometric interpretation of scaling the vector by 2?

  • What is the implicit line equation that goes through the points \((-4, -1, 0)\) and \((-1, 6, 0)\)?

  • Consider the same line as before. Use the implicit line equation to determine whether the point \((0, 0, 0)\) is above, below, or on the line.

  • Compute the unit vector of \(v_{ab}\) defined above.

  • Compute the dot product between \(v_{ab}\) and \(v_{ac}\).

  • Compute the cross product between \(v_{ab}\) and \((1,0,1)\).

2) Suppose a point p has barycentric coordinate (1,0,0) in relation to the triangle with points a, b, and c. What can we tell about the point p?

3) What is the difference between a vector and a point?

4) Suppose a point p has homogeneous coordinate \((2,-3,4,0)^T\). Is this point a point or a vector? What is the corresponding ordinary point of p?

5) Suppose a point p has 2D coordinate (2,4). What is the distance between p and the origin (0,0)?

6) Suppose we have a forward direction (-1,2,0) for an object. Construct an orthogonal basis for which (-1,2,0) corresopnds to the Z-axis with respect to the world coordinate frame.

Matrices and transforms

1) Suppose matrix A has dimentions 2x6 and matrix B has dimensions 3x2. Can we multiply AB? What about BA?

2) Suppose a 3x3 matrix A has an inverse \(A^{-1}\). What is \(A A^{-1}\)?

3) Suppose we are using XYZ euler angles. Give a 3x3 matrix expression that corresponds to the (30, 0, 90) degrees euler angles. Give your answer as a product of three 3x3 rotation matrices.

4) Give a 3x3 matrix that scales the height of an object by 2.

5) Give a 4x4 matrix that translates an object by (10, -5, 2).

6) Suppose we have an unit cube that is rotated by 30 degrees, scaled by 3, and then translated to (10,-5, 2). What will be the final position of the top right corner? Draw the effects of each transformation on the unit cube.

7) Consider again the unit cube. Suppose we translate it by (10,-5,2) and then rotate it by 30 degrees around the Z axis. What will be the final position of the top, right corner?

8) Consider the transformation from question 7, represent the series of transformations as a single transformation matrix in block matrix form.

9) Multiply the matrix \(\begin{bmatrix}3 & 2\\ 4 & -1 \end{bmatrix}\) and \(\begin{bmatrix}2 & 0\\ -2 & -1 \end{bmatrix}\)