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Matrix Transformations in 2D

対象: Universityカテゴリ: Math

説明

Visualizes how 2x2 matrices transform the plane by animating a NumberPlane grid through four key transformations: identity, 90-degree rotation, shear, and stretch. The basis vectors i-hat and j-hat are highlighted to show how the matrix columns determine the transformation outcome.

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Matrix Transformations in 2D

Description

Visualizes how 2x2 matrices transform the plane by animating a NumberPlane grid through four key transformations: identity, 90-degree rotation, shear, and stretch. The basis vectors i-hat and j-hat are highlighted to show how the matrix columns determine the transformation outcome.

Phases

# Phase Name Duration Description
1 Introduction 5s Display title, show NumberPlane with labeled basis vectors i-hat and j-hat
2 Identity Matrix 6s Show identity matrix MathTex, animate (no change), explain columns = basis images
3 Rotation 90° 8s Show rotation matrix, animate grid rotating 90°, basis vectors trace new positions
4 Shear Transform 8s Show shear matrix, animate horizontal shear, vertical lines tilt
5 Stretch Transform 8s Show stretch/scale matrix, animate axes-aligned scaling
6 Composition 10s Show two transforms composing, matrix multiplication = sequential transforms
7 Summary 5s Display all four matrices side by side with their geometric interpretations

Layout

+--------------------------------------------------+
|  Title: "Matrix Transformations in 2D"           |
+--------------------------------------------------+
|                                                  |
|   NumberPlane (center-left)  | Matrix MathTex   |
|   with grid lines            | (center-right)   |
|                              |                  |
|   i-hat (red arrow)          | [a  b]           |
|   j-hat (green arrow)        | [c  d]           |
|                              |                  |
|   Transformed grid           | Equation label   |
|   (animates via ApplyMatrix) |                  |
|                              |                  |
+--------------------------------------------------+
|  Bottom: formula / description text             |
+--------------------------------------------------+

Area Descriptions

  • Left 60%: NumberPlane with animated grid transformation
  • Right 40%: Matrix display (MathTex), transformation name, and formula
  • Bottom strip: Descriptive text about the current transformation

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • Use ApplyMatrix for grid transformation animations
  • Basis vectors should be drawn with Arrow objects and labeled with MathTex(r"\hat{i}") and MathTex(r"\hat{j}")
  • Reset the plane between each transformation using Transform back to identity
  • Color scheme: i-hat = RED, j-hat = GREEN, grid = BLUE tones, transformed = YELLOW