Study Guide 2 and 3

Quizzes are closed book. 30 minutes. In Lab. One written cheat sheet is allowed.

Topics:

Everything from study guide 1 as well as

  • Transformations

Terms To Know:

  • Frames of reference, coordinate systems

  • Homogeneous coordinates

  • Transformations: Scale, Rotate, and Translation

  • Skeletons, joints, bones, degrees of freedom, skinning, skin weights

Sample Short Questions:

  • Write the transformation that corresponds to a (0.5, 1, 0.5) scale

    • As a 2x2 block matrix

    • As a 4x4 matrix

  • Write the transformation that corresponds to a (0.5, 1, 0.5) scale

    • As a 2x2 block matrix

    • As a 4x4 matrix

  • Write the transformation that corresponds to a (45, 30, 15) degree XYZ Euler angle rotation

    • As a 2x2 block matrix

    • As a 4x4 matrix

  • What is the (x,y,z) coordinate that corresponds to the homogeneous point (1,2,3,4)?

  • What is the homogeneous coordinate for the vector (0,-1,0)?

  • What is the homogeneous coordinate for the point (0,-1,0)?

  • Suppose we rotate the vector (0,-1,0) by 30 degrees around the Z axis. What is its new direction?

  • Suppose we scale the vector (0,-1,0) by (0.5, 0.5, 0.5) units. What is its new length?

  • Suppose we translate the vector (0,-1,0) by (0.5, 0.5, 0.5) units. What is its new direction?

  • Suppose we rotate the point (0,-1,0) by 30 degrees around the Z axis. What is its new direction?

  • Suppose we scale the point (0,-1,0) by (0.5, 0.5, 0.5) units. What is its new length?

  • Suppose we translate the point (0,-1,0) by (0.5, 0.5, 0.5) units. What is its new position?

  • Give the rotation matrix that corresponds to the rotation given by the quaternion (x,y,z,w) = (sin(30), 0, 0, cos(30))

  • Suppose we want to rotate an object 30 degrees around the Y axis. Give this rotation as a

    • (x,y,z,w) quaternion

    • (x,y,z) euler angles (ZYX order)

  • Imagine a cylinder that extends from -1 to 1 along the Y axis and has origin (0,0,0) and radius 0.5 when unrotated, unscaled, and untranslated.

    • Draw the cylinder

    • What are the global coordinates of the top of the cylinder if we scale by 0.5, rotate 30 degrees, and translate by (-1,-1,-1)

      • Give the transform that corresponds to this sequence of transformations

      • Multiply (0,1,0) by this transform.

      • Show that the transform that translates by (1,1,0), rotates by -30 degrees, and scales by 2 is the inverse of the above matrix.

  • Suppose we have a transform that translates by (1,0,0) and rotates by 45 degrees. Give the transformation and inverse transformation corresponding to this sequence of transformations.

  • How can we use the cross product to check if a point is to the left or right of us?

  • Suppose we compute the cross product from (4,5,0) and (-1,5,0). In a left-handed coordinate system, what direction will the resulting vector be?

  • Consider the scene hierarchy in Unity. Suppose we have the following hierarchy: root → hips → knee → foot.

    • How can we compute from local foot coordinates to global coordinates?

    • What is the origin of the foot in local coordinates?

    • What is the up vector of the foot in local coordinates?

Sample Long questions can be found in our class notes and lab exercises.