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Quiz: 3D Geometry and Transformations

Test your understanding of 3D coordinate systems, rotations, and camera models.

1. A right-hand coordinate system is characterized by:

  1. All axes pointing to the right
  2. Curling right-hand fingers from X to Y points thumb in Z direction
  3. The Z-axis always pointing down
  4. Negative coordinates only
Show Answer

The correct answer is B. In a right-hand system, if you curl your right hand's fingers from the X-axis toward the Y-axis, your thumb points in the Z direction. This is the standard convention in mathematics and many graphics systems.

Concept Tested: 3D Coordinate System

2. Euler angles can suffer from gimbal lock when:

  1. All angles are zero
  2. Two rotation axes become aligned
  3. The angles are too small
  4. Only yaw is non-zero
Show Answer

The correct answer is B. Gimbal lock occurs when two rotation axes align (typically when pitch is ±90°), causing loss of one degree of freedom. This is an inherent limitation of Euler angle representation.

Concept Tested: Gimbal Lock

3. A quaternion representing rotation uses:

  1. Three components
  2. Four components with a unit norm constraint
  3. Nine components like a rotation matrix
  4. Two complex numbers
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The correct answer is B. A unit quaternion has four components \((w, x, y, z)\) with \(w^2 + x^2 + y^2 + z^2 = 1\). This provides a singularity-free representation of 3D rotations.

Concept Tested: Quaternion

4. Homogeneous coordinates allow:

  1. Only rotation operations
  2. Combining rotation and translation in a single matrix multiplication
  3. Eliminating the need for matrices
  4. Representing only 2D points
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The correct answer is B. Homogeneous coordinates add an extra dimension (e.g., \((x, y, z, 1)\) for 3D points), enabling both rotation and translation to be represented as matrix multiplication.

Concept Tested: Homogeneous Coordinates

5. An SE(3) transformation represents:

  1. Scaling only
  2. A rigid body motion (rotation and translation) in 3D
  3. Projection to 2D
  4. Color transformation
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The correct answer is B. SE(3) is the Special Euclidean group in 3D, representing all rigid body transformations—combinations of rotation and translation that preserve distances and angles.

Concept Tested: SE3 Transform

6. The camera intrinsic matrix \(K\) contains:

  1. The camera's position in the world
  2. Focal length and principal point (internal camera geometry)
  3. The rotation of the camera
  4. The image pixel values
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The correct answer is B. The intrinsic matrix contains internal camera parameters: focal lengths (\(f_x\), \(f_y\)), principal point (\(c_x\), \(c_y\)), and optionally skew. These describe how 3D camera coordinates project to 2D pixels.

Concept Tested: Intrinsic Parameters

7. Perspective projection causes:

  1. All objects to appear the same size
  2. Distant objects to appear smaller than near objects
  3. Parallel lines to remain parallel in the image
  4. Colors to change
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The correct answer is B. Perspective projection divides by depth (Z-coordinate), causing distant objects to appear smaller. This mimics human vision and camera optics.

Concept Tested: Perspective Projection

8. In stereo vision, depth is inversely proportional to:

  1. The baseline between cameras
  2. The disparity (difference in image positions)
  3. The focal length
  4. The image resolution
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The correct answer is B. The relationship \(Z = \frac{fb}{d}\) shows depth is inversely proportional to disparity. Larger disparity means closer objects; smaller disparity means farther objects.

Concept Tested: Stereo Vision

9. Triangulation in stereo vision:

  1. Detects triangular shapes
  2. Computes 3D position from corresponding 2D points in multiple views
  3. Measures triangle areas
  4. Filters noise from images
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The correct answer is B. Triangulation uses the intersection of projection rays from multiple cameras to determine the 3D position of a point observed in multiple images.

Concept Tested: Triangulation

10. A point cloud is:

  1. A type of weather pattern
  2. A set of 3D points representing surfaces or structure
  3. A 2D image format
  4. A neural network architecture
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The correct answer is B. A point cloud is a collection of 3D points, often with associated attributes (color, intensity, normals), representing the surface or structure of objects and environments.

Concept Tested: Point Cloud