How to Find Perspective Projection?

Finding perspective projection involves mapping a point from 3D space onto a 2D plane (or a line in a simplified case, as indicated by the reference) from a specific viewpoint. According to the provided reference, this transformation can be defined geometrically and calculated using a specific matrix formula.

Understanding Perspective Projection

At its core, perspective projection is a geometric transformation. It simulates how objects appear smaller the farther away they are, mimicking human vision or camera lenses.

• The process involves a viewpoint or center of perspectivity, denoted by v.
• It also involves a viewline (or viewplane in 3D), denoted by ℓ.

For any point p (that is not the viewpoint v), its perspective projection p′ onto the viewline ℓ is defined as the point where the line passing through the viewpoint v and the point p intersects the viewline ℓ. This is conceptually shown in diagrams illustrating projection rays converging at the viewpoint.

Calculating Perspective Projection Using the Reference Formula

While the geometric definition provides the intuition, the provided reference offers a specific method to calculate this projection. It states that the result of the perspective projection from v onto ℓ for a point p (where p ≠ v) is given by the expression:

p′ = vℓ⊤ − (ℓ · v)I3 p

This formula suggests the projection can be computed by applying a linear transformation (a matrix multiplication) to the point p, assuming v, ℓ, and p are represented as vectors and I3 is the 3x3 identity matrix.

Components of the Formula

To use this formula, you need to understand its components:

• v: Represents the viewpoint or center of perspectivity as a column vector (likely 3D, given I3).
• ℓ: Represents the viewline. Based on the operations (ℓ⊤ and ℓ · v), ℓ is treated as a column vector compatible with v. The specific meaning of ℓ as a vector representing a line is not fully detailed but is essential for the matrix calculations.
• ℓ⊤: The transpose of the vector ℓ, making it a row vector.
• vℓ⊤: The outer product of v and ℓ, resulting in a 3x3 matrix (assuming v and ℓ are 3x1 column vectors).
• ℓ · v: The dot product of ℓ and v, resulting in a scalar value.
• I3: The 3x3 identity matrix.
• p: Represents the point being projected as a column vector (likely 3D, like v).

Steps to Calculate Perspective Projection Using the Formula

Based on the formula p′ = vℓ⊤ − (ℓ · v)I3 p, here are the steps to find the projected point p′:

1. Represent Points and Line as Vectors: Define the viewpoint v, the viewline parameter ℓ (as a vector), and the point p as column vectors.
2. Calculate the Dot Product: Compute the scalar value s = ℓ · v.
3. Calculate the Outer Product: Compute the 3x3 matrix O = vℓ⊤.
4. Construct the Projection Matrix: Calculate the 3x3 projection matrix M = O - s * I3. This expands to M = vℓ⊤ − (ℓ · v)I3.
5. Apply the Matrix to the Point: Calculate the projected point p′ by multiplying the matrix M by the point vector p: p′ = M * p.

The resulting vector p′ is the perspective projection of point p onto the viewline ℓ from the viewpoint v, as defined by the provided reference formula.