In this section, we will explore the concepts of projections and views in 3D graphics. Understanding these concepts is crucial for rendering 3D objects onto a 2D screen. We will cover the following topics:
- Introduction to Projections
- Types of Projections
- Orthographic Projection
- Perspective Projection
- View Transformation
- Practical Examples and Exercises
- Introduction to Projections
Projections are methods used to represent three-dimensional objects on a two-dimensional plane (such as a computer screen). The main goal is to transform 3D coordinates into 2D coordinates while preserving the spatial relationships and visual appearance of the objects.
- Types of Projections
Orthographic Projection
Orthographic projection is a type of parallel projection where the projection lines are perpendicular to the projection plane. This type of projection does not account for perspective, meaning objects retain their size regardless of their distance from the camera.
Key Characteristics:
- Parallel lines remain parallel.
- No perspective distortion.
- Useful for technical drawings and CAD applications.
Example:
Consider a cube with vertices at the following coordinates:
To project this cube orthographically onto the XY plane, we simply ignore the Z coordinate:
Perspective Projection
Perspective projection simulates the way the human eye perceives the world. Objects that are farther away appear smaller, and parallel lines converge at a vanishing point.
Key Characteristics:
- Parallel lines converge.
- Objects shrink with distance.
- Provides a realistic view.
Example:
To project a point \((x, y, z)\) using perspective projection, we use the following formulas: \[ x' = \frac{x}{z} \] \[ y' = \frac{y}{z} \]
Here’s a simple Python function to perform perspective projection:
def perspective_projection(x, y, z):
# Assuming a simple perspective projection with focal length f = 1
f = 1
x_prime = f * (x / z)
y_prime = f * (y / z)
return x_prime, y_prime
# Example usage
point_3d = (2, 3, 4)
point_2d = perspective_projection(*point_3d)
print("Projected 2D point:", point_2d)
- View Transformation
View transformation involves transforming the coordinates of objects from the world coordinate system to the camera (or view) coordinate system. This process typically involves:
- Translation: Moving the entire scene so that the camera is at the origin.
- Rotation: Aligning the scene with the camera's orientation.
Example:
Consider a camera located at \((cx, cy, cz)\) and looking in a direction defined by a rotation matrix \(R\). The view transformation can be represented as: \[ V = R \cdot (P - C) \] where \(P\) is the point in world coordinates, \(C\) is the camera position, and \(V\) is the point in view coordinates.
- Practical Examples and Exercises
Example: Orthographic Projection of a Cube
Given a cube with vertices: \[ (1, 1, 1), (1, 1, -1), (1, -1, 1), (1, -1, -1), (-1, 1, 1), (-1, 1, -1), (-1, -1, 1), (-1, -1, -1) \]
Project the cube orthographically onto the XY plane.
Solution:
vertices_3d = [
(1, 1, 1), (1, 1, -1), (1, -1, 1), (1, -1, -1),
(-1, 1, 1), (-1, 1, -1), (-1, -1, 1), (-1, -1, -1)
]
vertices_2d = [(x, y) for x, y, z in vertices_3d]
print("Orthographic projection onto XY plane:", vertices_2d)Exercise: Perspective Projection of a Point
Write a function to perform perspective projection of a point \((x, y, z)\) with a given focal length \(f\).
Solution:
def perspective_projection(x, y, z, f):
x_prime = f * (x / z)
y_prime = f * (y / z)
return x_prime, y_prime
# Test the function
point_3d = (2, 3, 4)
focal_length = 1
point_2d = perspective_projection(*point_3d, focal_length)
print("Projected 2D point with focal length 1:", point_2d)Common Mistakes and Tips
- Ignoring the Z coordinate in perspective projection: Ensure you divide by the Z coordinate to achieve the correct perspective effect.
- Incorrect matrix multiplication order: When performing view transformations, the order of matrix multiplication matters. Always apply rotation before translation.
Conclusion
In this section, we covered the basics of projections and views, including orthographic and perspective projections, and view transformations. These concepts are fundamental for rendering 3D objects onto a 2D screen. Understanding these principles will help you create more realistic and accurate 3D graphics. Next, we will delve into surface modeling, where we will explore how to represent and manipulate the surfaces of 3D objects.
Mathematics 3D
Module 1: Fundamentals of Linear Algebra
- Vectors and Vector Spaces
- Matrices and Determinants
- Systems of Linear Equations
- Eigenvalues and Eigenvectors
Module 2: Linear Transformations
- Definition and Properties
- Transformation Matrices
- Rotations, Translations, and Scalings
- Composition of Transformations
Module 3: Geometry in 3D Space
- Coordinates and Planes
- Vectors in 3D Space
- Dot Product and Cross Product
- Equations of Planes and Lines