In computer graphics, rotation is the process of turning an object around a specified point or axis. The rotation can be performed in two dimensions (2D) or three dimensions (3D), and the object rotates by a specified angle, which can be positive or negative. 

In 2D graphics, rotation is typically performed around the origin of a coordinate system. This type of rotation changes the coordinates of each point in the object by a specified angle, using trigonometric functions like sine and cosine. There are two types of rotation in 2D graphics:

• Clockwise 
• Anti-clockwise 

1. Clockwise Rotation

The rotation angle corresponds to a negative angle.

Where,

• P is the original position
• P’ is the final position after rotation
• θ is the angle of rotation

Matrix for clockwise direction

2. Anti-clockwise Rotation

The anticlockwise rotation angle corresponds to a positive angle.

Where, 

• Q is the original position
• Q’ is the final position
• θ is the angle of rotation

Matrix for Anti-clockwise direction

Here are programs for the rotation of triangle and line in computer graphics:

Program for Rotation of a Triangle

#include <iostream>
#include <cmath>
using namespace std;

// Struct to define a 2D point
struct Point {
    int x, y;
};

void rotate(Point& pt, double angle) {
    double radians = angle * M_PI / 180.0;  // Convert angle to radians
    double cosValue = cos(radians);         // Calculate cosine of the angle
    double sinValue = sin(radians);         // Calculate sine of the angle

    // New coordinates after rotation
    int newX = pt.x * cosValue - pt.y * sinValue;
    int newY = pt.x * sinValue + pt.y * cosValue;

    pt.x = newX;  // Update point's x-coordinate
    pt.y = newY;  // Update point's y-coordinate
}

// Function to rotate all three points of a triangle
void rotateTriangle(Point& A, Point& B, Point& C, double angle) {
    rotate(A, angle);  // Rotate point A
    rotate(B, angle);  // Rotate point B
    rotate(C, angle);  // Rotate point C
}

int main() {
    // Define three points of a triangle
    Point A = {0, 0};
    Point B = {4, 0};
    Point C = {2, 3};

    double rotationAngle = 60.0;  // Define the rotation angle in degrees

    // Display points before rotation
    cout << "Before Rotation:" << endl;
    cout << "A: (" << A.x << ", " << A.y << ")" << endl;
    cout << "B: (" << B.x << ", " << B.y << ")" << endl;
    cout << "C: (" << C.x << ", " << C.y << ")" << endl;

    // Perform rotation on the triangle
    rotateTriangle(A, B, C, rotationAngle);

    // Display points after rotation
    cout << "After Rotation:" << endl;
    cout << "A: (" << A.x << ", " << A.y << ")" << endl;
    cout << "B: (" << B.x << ", " << B.y << ")" << endl;
    cout << "C: (" << C.x << ", " << C.y << ")" << endl;

    return 0;
}

Output

Before Rotation:
A: (0, 0)
B: (4, 0)
C: (2, 3)
After Rotation:
A: (0, 0)
B: (2, 3)
C: (-1, 3)

Program to Find Rotation of Line

#include <iostream>
#include <cmath>
using namespace std;

// Struct to represent a 2D point
struct Point {
    int x, y;
};

void rotateLine(Point& start, Point& end, double angle) {
    double radians = angle * M_PI / 180.0;  // Convert angle to radians
    double cosVal = cos(radians);           // Calculate cosine of the angle
    double sinVal = sin(radians);           // Calculate sine of the angle

    // Rotate the start point
    int newX1 = start.x * cosVal - start.y * sinVal;
    int newY1 = start.x * sinVal + start.y * cosVal;

    // Rotate the end point
    int newX2 = end.x * cosVal - end.y * sinVal;
    int newY2 = end.x * sinVal + end.y * cosVal;

    // Update the coordinates of both points
    start.x = newX1;
    start.y = newY1;
    end.x = newX2;
    end.y = newY2;
}

int main() {
    // Define two points representing the line
    Point start = {0, 0};
    Point end = {5, 5};

    double angle = 45.0;  // Rotation angle in degrees

    // Display the points before rotation
    cout << "Before rotation:" << endl;
    cout << "Start: (" << start.x << ", " << start.y << ")" << endl;
    cout << "End: (" << end.x << ", " << end.y << ")" << endl;

    // Rotate the line
    rotateLine(start, end, angle);

    // Display the points after rotation
    cout << "After rotation:" << endl;
    cout << "Start: (" << start.x << ", " << start.y << ")" << endl;
    cout << "End: (" << end.x << ", " << end.y << ")" << endl;

    return 0;
}

Output

Before rotation:
Start: (0, 0)
End: (5, 5)
After rotation:
Start: (0, 0)
End: (0, 7)

In 3D computer graphics, objects are rotated about a fixed axis (x-axis, y-axis, or z-axis) rather than a point. There are three basic types of 3D rotations:

• Rotation around the X-axis
• Rotation around the Y-axis
• Rotation around the Z-axis

1. Rotation around the X-axis

 A point (x, y, z) rotates around the X-axis:

2. Rotation around the Y-axis

 A point (x, y, z) rotates around the Y-axis:

3. Rotation around the Z-axis

A point (x, y, z) rotates around the Z-axis:

In conclusion, rotation is an essential operation in computer graphics that allows for the manipulation and transformation of objects in both 2D and 3D spaces. Understanding the mathematical principles behind rotation, such as the use of rotation matrices and coordinates are critical for working with computer graphics.

1. What is a rotation matrix?

A rotation matrix is a mathematical tool used to rotate points or objects in space. It is a matrix that, when multiplied by the coordinates of a point, rotates the point by a specified angle.

2. What are homogeneous coordinates in rotation? 

Homogeneous coordinates extend the 2D or 3D coordinate system to a higher dimension, allowing for more complex transformations, including rotation, translation, and scaling, all in one unified framework.