Projecting Points onto Planes in 3D Space: Mathematical Principles and Code Implementation

Introduction

In three-dimensional geometry, projecting a point onto a plane is a fundamental operation with applications in computer graphics, physics engines, and engineering simulations. Given a 3D point and a plane defined by an origin point and a unit normal vector, this paper presents a vector-based projection method. The core idea is to calculate the perpendicular distance from the point to the plane along the normal direction and derive the projected point through vector operations.

Mathematical Foundation

Let point P have coordinates (point_x, point_y, point_z), and the plane be defined by origin O(orig_x, orig_y, orig_z) and unit normal vector N(normal_dx, normal_dy, normal_dz). First, construct vector V from O to P:

V = P - O

Compute the dot product of vector V and normal vector N to obtain the signed distance dist from P to the plane:

dist = V · N = Vx * Nx + Vy * Ny + Vz * Nz

Here, dist represents the scalar distance from point P to the plane along the normal N. Finally, subtract the vector dist * N from point P to get the projected point P_proj:

P_proj = P - dist * N

This method ensures P_proj lies on the plane, as moving from P along the normal direction by distance dist precisely reaches the plane. Combined with graphical illustrations, the red vector is V, blue and green vectors represent dist * N, and subtracting the green vector achieves projection.

Code Implementation

The following code examples implement the projection algorithm in C++ and C, assuming vector structures and necessary mathematical functions are defined.

In C++, using a custom vector class or standard libraries:

#include <iostream>
#include <cmath>

struct Vector3 {
    double x, y, z;
    Vector3(double x = 0, double y = 0, double z = 0) : x(x), y(y), z(z) {}
    Vector3 operator-(const Vector3& other) const {
        return Vector3(x - other.x, y - other.y, z - other.z);
    }
    double dot(const Vector3& other) const {
        return x * other.x + y * other.y + z * other.z;
    }
    Vector3 operator*(double scalar) const {
        return Vector3(x * scalar, y * scalar, z * scalar);
    }
};

Vector3 projectPointToPlane(const Vector3& point, const Vector3& origin, const Vector3& normal) {
    Vector3 v = point - origin;
    double dist = v.dot(normal);  // Assuming normal is a unit vector
    Vector3 projected = point - normal * dist;
    return projected;
}

int main() {
    Vector3 point(10.0, 20.0, -5.0);
    Vector3 origin(0.0, 10.0, 0.0);
    Vector3 normal(0.0, 1.0, 0.0);  // Unit normal vector
    Vector3 result = projectPointToPlane(point, origin, normal);
    std::cout << "Projected point: (" << result.x << ", " << result.y << ", " << result.z << ")" << std::endl;
    return 0;
}

In C, defining similar structures and functions:

#include <stdio.h>
#include <math.h>

typedef struct {
    double x, y, z;
} Vector3;

Vector3 vector_subtract(Vector3 a, Vector3 b) {
    Vector3 result = {a.x - b.x, a.y - b.y, a.z - b.z};
    return result;
}

double vector_dot(Vector3 a, Vector3 b) {
    return a.x * b.x + a.y * b.y + a.z * b.z;
}

Vector3 vector_scale(Vector3 v, double scalar) {
    Vector3 result = {v.x * scalar, v.y * scalar, v.z * scalar};
    return result;
}

Vector3 project_point_to_plane(Vector3 point, Vector3 origin, Vector3 normal) {
    Vector3 v = vector_subtract(point, origin);
    double dist = vector_dot(v, normal);  // Assuming normal is a unit vector
    Vector3 offset = vector_scale(normal, dist);
    Vector3 projected = vector_subtract(point, offset);
    return projected;
}

int main() {
    Vector3 point = {10.0, 20.0, -5.0};
    Vector3 origin = {0.0, 10.0, 0.0};
    Vector3 normal = {0.0, 1.0, 0.0};  // Unit normal vector
    Vector3 result = project_point_to_plane(point, origin, normal);
    printf("Projected point: (%f, %f, %f)", result.x, result.y, result.z);
    return 0;
}

Supplementary Information

As a reference, an alternative method uses the plane equation. A plane can be represented as Ax + By + Cz + d = 0, where (A, B, C) is the normal vector and d is a constant. The distance from point P to the plane can be computed with the formula |Ax + By + Cz + d| / sqrt(A^2 + B^2 + C^2), but this paper focuses on the vector-based approach for its directness and ease of implementation.

Conclusion

Through vector algebra and simple operations, points can be efficiently projected onto planes in 3D space. The provided code examples demonstrate how to implement this algorithm in C++ and C, suitable for various engineering and scientific applications. Ensuring the normal vector is unit length is critical to avoid scaling errors.