Mathematics of Computer Graphics

Larry Riddle, May 16, 2018

These apps were developed to be used in conjunction with the textbook "An Introduction to the Mathematics of Computer Graphics" by Nathan Carter, Bentley University.

Table of Contents

  1. Affine Transformations
    1. 2d Affine Transformations
    2. 3d Affine Transformations
  2. Lines of Sight
    1. Line of Sight Demo
    2. Transformation Principle
  3. Three Color Models
    1. RGB Color Model
    2. CMY Color Model
    3. RGB/HSV Color Converter
  4. Bezier Curves
    1. Cubic Bezier Curve via Vectors
    2. Cubic Bezier Curve via Control Points
    3. 3d Cubic Bezier Curves
    4. General Bezier Curves
    5. de Casteljau's algorithm
    6. Bernstein Polynomials
    7. Continuity for Bezier Curves

1.

Affine Transformations

  • 1. 2d Affine Transformations
  • 2. 3d Affine Transformations

1.1.

2d Affine Transformations

Click on the S, T, or R buttons to scale, translate, or rotate. Specify the values by using the sliders or entering the values in the text boxes. For S and T, the top box (slider) is for the x direction and the bottom box (slider) is for the y direction. Check the Lock box to scale by the same amount in both the x and y directions.[br]You can move the colored corner points with the mouse to change the original shape.[br]The Restore button sets all values back to their default values.[br]The Reset button deletes all transformations.[br]Copy Image will make a copy of the transformed image at its present location. Use Delete Copy to delete this.[br]Clear All will reset everything back to the original settings.[br]Check Hide Original Shape to show only the transformed image.
– Larry Riddle

1.2.

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2.

Lines of Sight

  • 1. Line of Sight Demo
  • 2. Transformation Principle

2.1.

Line of Sight Demo

* Use the Camera slider to move the location of the camara.[br]* Use the mouse to move the pixel in the rendering window. The pixel will turn red when the line of sight intersects the sphere.[br]* If you check "Trace On" and move the pixel, you will get a simulation of the image of the sphere being drawn on the rendering window. Use the arrow keys to scan the pixel horizontally or vertically across the rending window.[br]* Move the sphere by moving the center with the mouse or entering the coordinates in the textbox. The radius can be changed via the textbox.[br]* The size of the rendering window can be changed with the slider.[br][br](Based on original concept by Kevin Cox, Bentley University)
– Larry Riddle

2.2.

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3.

Three Color Models

  • 1. RGB Color Model
  • 2. CMY Color Model
  • 3. RGB/HSV Color Converter

3.1.

RGB Color Model

Turn on the colors by checking the appropriate box. The sliders are used to change the intensity of the three primary colors.[br][br]Circles cannot be moved when visible if they are on top of one another. Uncheck a box color to move one of the other circles.
– Larry Riddle

3.2.

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3.3.

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4.

Bezier Curves

  • 1. Cubic Bezier Curve via Vectors
  • 2. Cubic Bezier Curve via Control Points
  • 3. 3d Cubic Bezier Curves
  • 4. General Bezier Curves
  • 5. de Casteljau's algorithm
  • 6. Bernstein Polynomials
  • 7. Continuity for Bezier Curves

4.1.

Cubic Bezier Curve via Vectors

Use the mouse or enter the starting and ending points, P1 and P2, for the Bezier curve. Move points A and B to change the initial and final vectors, or enter the coordinates of A = P1 + v1 and B = P2 + v2.
– Larry Riddle

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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Information