3D Coordinate Systems, Direction Cosines, Ratios & Projections

1. The Right-Handed Cartesian Frame $\mathbb{R}^3$

In three-dimensional Euclidean space $\mathbb{R}^3$, three mutually perpendicular oriented lines intersecting at a common origin $O(0, 0, 0)$ define the coordinate axes: the $x$-axis, $y$-axis, and $z$-axis. By standard convention, these axes satisfy the Right-Hand Rule: rotating the positive $x$-axis into the positive $y$-axis through $\pi/2$ advances a right-handed screw along the positive $z$-axis ($\hat{i} \times \hat{j} = \hat{k}$).

The three coordinate planes partition space into eight octants:

Any point $P \in \mathbb{R}^3$ is uniquely identified by the ordered triplet of signed perpendicular distances $(x, y, z)$.

2. Cylindrical Coordinates $(\rho, \phi, z)$

Cylindrical coordinates combine 2D polar coordinates in the $xy$-plane with the Cartesian altitude $z$:

where $\rho = \sqrt{x^2 + y^2} \ge 0$ is the radial distance from the $z$-axis, and $\phi \in [0, 2\pi)$ is the azimuthal angle measured counterclockwise from the positive $x$-axis. The differential volume element is:

3. Spherical Polar Coordinates $(r, \theta, \phi)$

Spherical coordinates specify the Euclidean distance $r$ from the origin, the polar/colatitude angle $\theta$ from the positive $z$-axis, and the azimuthal angle $\phi$ in the $xy$-plane:

The Jacobian determinant of this transformation yields the spherical volume element:

1. Euclidean Distance Metric in $\mathbb{R}^3$

Let $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ be two distinct points in space. By double application of the Pythagorean theorem across the rectangular box having $P_1 P_2$ as main spatial diagonal:

2. Section Formulas (Internal & External Division)

Let $P(x, y, z)$ divide the line segment joining $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in the ratio $m : n$:

• Internal Division ($m : n > 0$): $$\mathbf{P = \left( \frac{m x_2 + n x_1}{m + n}, \, \frac{m y_2 + n y_1}{m + n}, \, \frac{m z_2 + n z_1}{m + n} \right)}$$
• External Division ($m : -n$): $$\mathbf{P = \left( \frac{m x_2 - n x_1}{m - n}, \, \frac{m y_2 - n y_1}{m - n}, \, \frac{m z_2 - n z_1}{m - n} \right)} \quad (m \ne n)$$

3. Centroids of Spatial Polygons and Polyhedra

• Triangle Centroid: For vertices $A, B, C$, the centroid $G$ is the concurrency point of the three medians: $$G = \left( \frac{x_A + x_B + x_C}{3}, \, \frac{y_A + y_B + y_C}{3}, \, \frac{z_A + z_B + z_C}{3} \right)$$
• Tetrahedron Centroid: For vertices $A, B, C, D$, the centroid $G$ divides each line segment joining a vertex to the centroid of the opposite face in the ratio $3 : 1$: $$G = \left( \frac{x_A + x_B + x_C + x_D}{4}, \, \frac{y_A + y_B + y_C + y_D}{4}, \, \frac{z_A + z_B + z_C + z_D}{4} \right)$$

1. Direction Angles and Direction Cosines

Let an oriented ray $L$ pass through the origin $O$ in direction of unit vector $\hat{u}$. Let $\alpha, \beta, \gamma \in [0, \pi]$ be the positive inclination angles made by $L$ with the positive $x$, $y$, and $z$ axes respectively. These are the direction angles of $L$.

Their cosines are called the Direction Cosines (DCs), conventionally denoted by $(l, m, n)$:

2. Rigorous Proof of the Fundamental Identity: $l^2 + m^2 + n^2 = 1$

Let $P(x, y, z)$ be a point on line $L$ at distance $r = \sqrt{x^2 + y^2 + z^2} > 0$ from origin $O$. Projecting $P$ orthogonally onto the coordinate axes:

Squaring and adding these three projection relations:

Since $x^2 + y^2 + z^2 = r^2$, dividing both sides by $r^2 \ne 0$ establishes:

Consequently, the unit direction vector along the line is identically $\hat{u} = l\hat{i} + m\hat{j} + n\hat{k}$.

3. Direction Ratios (DRs) & Normalization

Any three real numbers $(a, b, c)$ proportional to the direction cosines $(l, m, n)$ are called Direction Ratios (DRs):

Substituting into $l^2 + m^2 + n^2 = 1$:

1. Projection of a Line Segment onto an Oriented Line

Let $AB$ be a directed line segment with initial point $A(x_1, y_1, z_1)$ and terminal point $B(x_2, y_2, z_2)$. Let $L$ be an oriented line with direction cosines $(l, m, n)$.

The vector displacement is $\vec{AB} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$. The orthogonal projection $p$ of $AB$ onto line $L$ is the scalar dot product with the unit direction vector $\hat{u} = l\hat{i} + m\hat{j} + n\hat{k}$:

2. Length of a Line Segment in Terms of Projections

Projecting segment $AB$ onto the three orthogonal coordinate axes gives projections $p_x = x_2 - x_1$, $p_y = y_2 - y_1$, and $p_z = z_2 - z_1$. The length of $AB$ is the Euclidean norm of its projections:

Furthermore, if $p_1, p_2, p_3$ are projections of $AB$ onto any three mutually orthogonal spatial lines, then $AB^2 = p_1^2 + p_2^2 + p_3^2$.

1. Angle Between Two Lines

Let $L_1$ and $L_2$ be two lines with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ and unit vectors $\hat{u}_1, \hat{u}_2$. The angle $\theta \in [0, \pi]$ between them is given by:

Using Lagrange's trigonometric identity, $\sin^2\theta = 1 - \cos^2\theta = (l_1^2 + m_1^2 + n_1^2)(l_2^2 + m_2^2 + n_2^2) - (l_1 l_2 + m_1 m_2 + n_1 n_2)^2$:

• Orthogonality Criterion ($L_1 \perp L_2$): $$\mathbf{l_1 l_2 + m_1 m_2 + n_1 n_2 = 0 \quad \Longleftrightarrow \quad a_1 a_2 + b_1 b_2 + c_1 c_2 = 0}$$
• Parallelism Criterion ($L_1 \parallel L_2$): $$\mathbf{\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2} \quad \Longleftrightarrow \quad \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}}$$

2. Perpendicular Distance of a Point from a Line

Let $P(x_1, y_1, z_1)$ be a point in space, and let line $L$ pass through $A(x_0, y_0, z_0)$ with direction cosines $(l, m, n)$. Let $M$ be the foot of the perpendicular from $P$ onto $L$. In right triangle $\triangle APM$: