Projection Formula in Three-Dimensional Cartesian Coordinate System

Introduction

In three-dimensional (3D) Cartesian coordinate systems, projections play a critical role in reducing the dimensionality of spatial data while preserving essential geometric properties. Projections allow us to map points, lines, and planes onto lower-dimensional spaces, facilitating calculations in geometry, physics, and computer graphics. The projection formula provides a systematic method for determining the shadow or image of a point or object when viewed from a specific direction. This article explores the mathematical foundations of projections in 3D space, focusing on the orthogonal and oblique projection formulas, their derivations, and applications.

Mathematical Foundations

A three-dimensional Cartesian coordinate system consists of three perpendicular axes—x, y, and z—intersecting at the origin (0, 0, 0). Any point in space can be represented as $ \mathbf{P} = (x, y, z) $, where $ x, y, z $ are real numbers. Projections involve decomposing a vector or point into components along a specific direction, often referred to as the projection axis. The projection formula relies on vector algebra, including dot products and cross products, to determine the relationship between vectors and their projections.

The orthogonal projection of a vector $ \mathbf{v} $ onto another vector $ \mathbf{u} $ is the component of $ \mathbf{v} $ that lies in the direction of $ \mathbf{u} $. This projection is calculated using the dot product:
$$ \text{proj}_{\mathbf{u}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{u}|^2 \right) \mathbf{u} $$
Here, $ \mathbf{v} \cdot \mathbf{u} $ is the dot product, and $ |\mathbf{u}| $ is the magnitude of $ \mathbf{u} $. This formula ensures that the projection is scaled to align with the direction of $ \mathbf{u} $.

For oblique projections, the projection direction is not orthogonal to the projection plane, introducing a scalar factor to account for the angle between the projection axis and the plane. The formula for an oblique projection of $ \mathbf{v} $ onto a plane with normal vector $ \mathbf{n} $ is:
$$ \text{proj}_{\text{plane}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{n}}{|\mathbf{n}|^2 \right) \mathbf{n} $$
This formula adjusts the projection by considering the angle between the vector and the plane, allowing for non-orthogonal projections.

Orthogonal Projection Formula

The orthogonal projection formula is pivotal in 3D geometry, as it enables the calculation of shadows or projections of points onto planes or axes. For instance, projecting a point $ \mathbf{P} = (x, y, z) $ onto the x-axis involves ignoring the y and z components, resulting in $ \text{proj}_{x} \mathbf{P} = (x, 0, 0) $. Similarly, projecting onto the y-axis yields $ (0, y, 0) $, and onto the z-axis gives $ (0, 0, z) $.

Generalizing this to arbitrary planes, the orthogonal projection of a point $ \mathbf{P} $ onto a plane with normal vector $ \mathbf{n} $ requires solving the equation $ \mathbf{P} - \text{proj}{\mathbf{n}} \mathbf{P} = \mathbf{0} $, where $ \text{proj}{\mathbf{n}} \mathbf{P} $ is the projection of $ \mathbf{P} $ onto the plane. This projection is determined by subtracting the component of $ \mathbf{P} $ along $ \mathbf{n} $ from $ \mathbf{P} $ itself.

Oblique Projection Formula

Oblique projections introduce a scalar factor to the orthogonal projection formula, reflecting the angle between the projection axis and the plane. For example, projecting a vector $ \mathbf{v} $ onto a plane with normal vector $ \mathbf{n} $ involves scaling the orthogonal projection by a factor $ \cos\theta $, where $ \theta $ is the angle between $ \mathbf{v} $ and $ \mathbf{n} $. The formula becomes:
$$ \text{proj}_{\text{plane}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{n}}{|\mathbf{n}|^2 \right) \mathbf{n} $$
This adjustment ensures that the projection accounts for the inclination of the vector relative to the plane, making it applicable in scenarios where non-orthogonal projections are necessary.

Projection of a Point onto a Plane

To project a point $ \mathbf{P} $ onto a plane defined by the equation $ \mathbf{a} \cdot \mathbf{r} = d $, where $ \mathbf{a} $ is the normal vector and $ \mathbf{r} $ is a position vector, we solve for the closest point on the plane to $ \mathbf{P} $. The projection formula is derived by minimizing the distance between $ \mathbf{P} $ and a general point $ \mathbf{r} $ on the plane. This leads to the solution:
$$ \mathbf{Q} = \mathbf{P} - \left( \frac{\mathbf{a} \cdot (\mathbf{P} - \mathbf{a})}{|\mathbf{a}|^2} \right) \mathbf{a} $$
Here, $ \mathbf{Q