1. Introduction

Definition: Displaying 2D geometric objects involves drawing basic shapes such as points, lines, circles, and polygons on a 2D plane using computer graphics systems. These are fundamental building blocks for graphics applications like CAD, UI design, plotting, and games.

Key Concept: 2D graphics rely on a coordinate system, typically the Cartesian $(x, y)$ plane, to position and render shapes.

2. Coordinate Systems in 2D Graphics

1. World Coordinates:

   • Defines positions relative to a logical scene
   • Independent of the screen or device resolution

2. Device Coordinates:

   • Maps world coordinates to actual screen pixels
   • Conversion from world → device coordinates is necessary for rendering

3. Viewport:

   • Specifies the rectangular portion of the window where objects are drawn

3. Basic 2D Geometric Primitives

A. Point

• Definition: The simplest object in 2D graphics, defined by a single coordinate $(x, y)$.
• Usage: Plotting data, vertices of polygons, or markers.

B. Line

• Definition: A straight connection between two points $(x_1, y_1)$ and $(x_2, y_2)$.

• Drawing Algorithms:

  1. DDA (Digital Differential Analyzer): Incremental algorithm using floating-point arithmetic.
  2. Bresenham’s Line Algorithm: Efficient integer-based algorithm for raster devices.

C. Circle

• Definition: Set of points equidistant from a center $(x_c, y_c)$ with radius $r$.

• Equation: $(x - x_c)^2 + (y - y_c)^2 = r^2$

• Drawing Algorithms:

  • Midpoint Circle Algorithm
  • Bresenham’s Circle Algorithm

D. Ellipse

• Definition: Set of points where the sum of distances from two foci is constant.
• Equation: $\frac{(x - x_c)^2}{a^2} + \frac{(y - y_c)^2}{b^2} = 1$
• Drawing Algorithm: Midpoint Ellipse Algorithm

E. Polygon

• Definition: Closed shape made of lines connecting a sequence of points.

• Types: Triangles, rectangles, pentagons, etc.

• Filling Algorithms:

  • Scanline fill algorithm
  • Flood-fill algorithm

4. Transformations of 2D Objects

To manipulate 2D shapes, several transformations are applied:

1. Translation: Move object by $(dx, dy)$

2. Scaling: Resize object relative to origin or a reference point

3. Rotation: Rotate object around the origin by angle $\theta$

4. Reflection: Mirror object along x-axis, y-axis, or a line

5. Shearing: Skew object along x or y-axis

5. Rendering 2D Objects

Steps to Display:

1. Convert world coordinates to device coordinates.
2. Apply transformation matrices if needed.
3. Use a rasterization algorithm to generate pixels.
4. Store pixels in the framebuffer and display on the screen.

Example in OpenGL (Pseudocode):

// Draw a triangle
glBegin(GL_TRIANGLES);
   glVertex2f(0.0, 0.0);
   glVertex2f(0.5, 0.0);
   glVertex2f(0.25, 0.5);
glEnd();
glFlush();

6. Applications

• User Interface Elements: Buttons, sliders, icons
• Computer-Aided Design (CAD): Basic drafting and modeling
• Graph Plotting: Charts, graphs, and scientific visualization
• Games: Sprites, backgrounds, and simple 2D animations

Key Takeaways

• 2D geometric objects are the building blocks of computer graphics.
• Efficient algorithms are used for drawing primitives like lines, circles, and polygons.
• Transformations allow manipulation of objects for animation, design, and interaction.
• Mastering 2D graphics is essential before moving to 3D graphics and rendering pipelines.