Cartesian & Polar Coordinate Systems, Distances & Transformations

1. The Cartesian Coordinate System & Metric Geometry

The foundation of analytic geometry, pioneered by René Descartes and Pierre de Fermat, establishes a bijective correspondence between the Euclidean plane $\mathbb{E}^2$ and the Cartesian product of the real field $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$. Any point $P \in \mathbb{E}^2$ is uniquely identified by an ordered pair of real coordinates $(x, y)$, representing signed perpendicular distances from two mutually orthogonal directed axes: the horizontal abscissa ($x$-axis) and the vertical ordinate ($y$-axis).

Under the standard Euclidean metric tensor $g_{ij} = \delta_{ij}$, the fundamental distance $d(P_1, P_2)$ between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is established directly via the Pythagorean theorem: $$d(P_1, P_2) = \|\mathbf{r}_2 - \mathbf{r}_1\|_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This Euclidean metric satisfies the three formal metric space axioms:

• Positive Definiteness: $d(P_1, P_2) \ge 0$, with $d(P_1, P_2) = 0 \iff P_1 = P_2$.
• Symmetry: $d(P_1, P_2) = d(P_2, P_1)$ for all $P_1, P_2 \in \mathbb{R}^2$.
• Triangle Inequality: $d(P_1, P_3) \le d(P_1, P_2) + d(P_2, P_3)$, with equality holding if and only if $P_2$ lies on the straight segment $\overline{P_1 P_3}$.

2. The General Section Formula (Internal and External Division)

Let $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ be two distinct points in $\mathbb{R}^2$. A point $P(x, y)$ dividing the directed line segment $P_1 P_2$ in the ratio $m : n$ satisfies the vector relationship: $$\frac{\vec{P_1 P}}{\vec{P P_2}} = \frac{m}{n} \iff n(\mathbf{r} - \mathbf{r}_1) = m(\mathbf{r}_2 - \mathbf{r})$$ Solving for the position vector $\mathbf{r} = (x, y)$: $$\mathbf{r} = \frac{m \mathbf{r}_2 + n \mathbf{r}_1}{m + n}$$ In scalar Cartesian components: $$x = \frac{m x_2 + n x_1}{m + n}, \qquad y = \frac{m y_2 + n y_1}{m + n}$$

Classification of Division:

• Internal Division ($m/n > 0$): The point $P$ lies strictly between $P_1$ and $P_2$. When $m = n = 1$, $P$ is the midpoint: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\\right)$$
• External Division ($m/n < 0$, with $m \ne -n$): Setting the ratio as $m : -n$, the point $P_{\text{ext}}$ lies on the extension of line segment $P_1 P_2$: $$x_{\text{ext}} = \frac{m x_2 - n x_1}{m - n}, \qquad y_{\text{ext}} = \frac{m y_2 - n y_1}{m - n}$$
• Harmonic Conjugates: The points $P_{\text{int}}$ and $P_{\text{ext}}$ dividing $P_1 P_2$ internally and externally in the same absolute ratio $m:n$ form a harmonic range $(P_1, P_2; P_{\text{int}}, P_{\text{ext}}) = -1$, meaning their distances satisfy the classical harmonic mean relation: $$\frac{2}{P_1 P_2} = \frac{1}{P_1 P_{\text{int}}} + \frac{1}{P_1 P_{\text{ext}}}$$

3. Classic Triangle Centers in the Cartesian Plane

For a triangle $\triangle ABC$ with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ and opposite side lengths $a = BC, b = CA, c = AB$:

• Centroid ($G$): The concurrence point of the three medians, dividing each median in ratio $2:1$: $$G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\\right)$$
• Incenter ($I$): The center of the inscribed circle, concurrence of internal angle bisectors: $$I = \left(\frac{a x_1 + b x_2 + c x_3}{a + b + c}, \frac{a y_1 + b y_2 + c y_3}{a + b + c}\\right)$$
• Excenters ($I_a, I_b, I_c$): Centers of the three excircles. The excenter opposite to vertex $A$ is: $$I_a = \left(\frac{-a x_1 + b x_2 + c x_3}{-a + b + c}, \frac{-a y_1 + b y_2 + c y_3}{-a + b + c}\\right)$$
• Euler Line Theorem: In any non-equilateral triangle, the orthocenter $H$, centroid $G$, and circumcenter $O$ are strictly collinear, satisfying the constant harmonic segment ratio: $$OG : GH = 1 : 2 \iff \mathbf{r}_H = 3\mathbf{r}_G - 2\mathbf{r}_O$$

1. Determinant Formulation for the Area of a Triangle

Consider a triangle $\triangle ABC$ formed by non-collinear vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ arranged counterclockwise. The signed area $\mathcal{A}$ is given by the cross product of the edge vectors $\vec{AB} = (x_2 - x_1, y_2 - y_1)$ and $\vec{AC} = (x_3 - x_1, y_3 - y_1)$: $$\mathcal{A} = \frac{1}{2} \left[ (x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1) \\right]$$ Expanding this algebraic expression yields the celebrated $3 \times 3$ determinant formulation: $$\mathcal{A} = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \frac{1}{2} \left[ x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \\right]$$ The physical unsigned area is given by the absolute value $|\mathcal{A}|$.

2. The Exact Criterion for Collinearity

Three distinct points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ lie on a common straight line if and only if the triangle they span degenerates into a segment with zero area: $$\text{Collinear} \iff \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0 \iff x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0$$ Alternatively, collinearity is characterized by equal slopes: $$m_{AB} = m_{BC} \iff \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2} \quad (x_1 \ne x_2 \ne x_3)$$

3. The Shoelace Formula for General Simple Polygons

By applying Green's Theorem $\iint_D dA = \frac{1}{2} \oint_{\partial D} (x \, dy - y \, dx)$ to a planar polygonal domain bounded by $n$ ordered vertices $P_1(x_1, y_1), P_2(x_2, y_2), \dots, P_n(x_n, y_n)$ traversed counterclockwise, the exact area is given by the Shoelace Formula: $$\mathcal{A}_n = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \\right| = \frac{1}{2} \left| (x_1 y_2 + x_2 y_3 + \dots + x_n y_1) - (y_1 x_2 + y_2 x_3 + \dots + y_n x_1) \\right|$$ where cyclic indexing applies such that $(x_{n+1}, y_{n+1}) \equiv (x_1, y_1)$.

1. Coordinate Definition & Bijective Transitions

In the polar coordinate system, a point $P$ in the Euclidean plane is specified relative to a fixed origin $O$ (termed the pole) and a horizontal directed ray extending to the right (the polar axis). The point is denoted by the ordered pair $(r, \theta)$:

• Radial Coordinate ($r$): The directed Euclidean distance from pole $O$ to point $P$, $r \in [0, \infty)$.
• Angular Coordinate ($\theta$): The counterclockwise angle measured from the polar axis to the ray $OP$, $\theta \in (-\pi, \pi]$ or $\theta \in [0, 2\pi)$.

The exact bijective conversion between Cartesian $(x, y)$ and Polar $(r, \theta)$ representations is given by: $$\begin{cases} x = r \cos \theta \\ y = r \sin \theta \end{cases} \iff \begin{cases} r = \sqrt{x^2 + y^2} \\ \theta = \operatorname{atan2}(y, x) \end{cases}$$ where $\operatorname{atan2}(y, x)$ resolves the proper quadrant of $\theta$ without ambiguity: $$\operatorname{atan2}(y, x) = \begin{cases} \arctan(y/x) & x > 0 \\ \arctan(y/x) + \pi & x < 0, \, y \ge 0 \\ \arctan(y/x) - \pi & x < 0, \, y < 0 \\ +\pi/2 & x = 0, \, y > 0 \\ -\pi/2 & x = 0, \, y < 0 \\ \text{undefined} & x = 0, \, y = 0 \end{cases}$$

2. Distance Formula & Triangle Area in Polar Form

Let $P_1(r_1, \theta_1)$ and $P_2(r_2, \theta_2)$ be two points expressed in polar coordinates. In $\triangle O P_1 P_2$, the angle subtended at the pole is $|\theta_2 - \theta_1|$. By the Law of Cosines: $$d(P_1, P_2)^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_2 - \theta_1)$$ $$d(P_1, P_2) = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_2 - \theta_1)}$$

The area of the triangle $\triangle O P_1 P_2$ formed by the pole and the two points is: $$\mathcal{A}_{\triangle O P_1 P_2} = \frac{1}{2} r_1 r_2 \sin|\theta_2 - \theta_1|$$ For three arbitrary points $P_1(r_1, \theta_1), P_2(r_2, \theta_2), P_3(r_3, \theta_3)$, the enclosed area is: $$\mathcal{A} = \frac{1}{2} \left| r_1 r_2 \sin(\theta_2 - \theta_1) + r_2 r_3 \sin(\theta_3 - \theta_2) + r_3 r_1 \sin(\theta_1 - \theta_3) \\right|$$

1. Algebraic Mechanics of Axis Translation

In many analytical geometric investigations, the mathematical form of a curve simplifies dramatically when the origin of coordinates is shifted to a new point $O'(h, k)$ while maintaining the parallel orientation and direction of the axes.

Let $(x, y)$ denote the coordinates of a point $P$ referred to the original axes $Ox, Oy$, and let $(X, Y)$ denote the coordinates of the same physical point $P$ referred to the translated axes $O'X, O'Y$. From vector addition: $$\mathbf{r} = \mathbf{r}_{O'} + \mathbf{r}' \implies \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} X + h \\ Y + k \end{pmatrix}$$ Conversely, the new coordinates in terms of the old are: $$\begin{cases} X = x - h \\ Y = y - k \end{cases}$$

2. Transformation of Algebraic Curves

Given a planar curve described by the implicit polynomial equation $f(x, y) = 0$, its transformed equation in the new coordinate frame $(X, Y)$ is obtained by direct substitution: $$F(X, Y) = f(X + h, Y + k) = 0$$

Invariance Under Pure Translation:

• Distance Invariance: $d(P_1, P_2) = \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Slope Invariance: $M = \frac{Y_2 - Y_1}{X_2 - X_1} = \frac{(y_2 - k) - (y_1 - k)}{(x_2 - h) - (x_1 - h)} = \frac{y_2 - y_1}{x_2 - x_1} = m$.
• Angle & Area Invariance: Translation preserves all angles between intersecting curves and all enclosed polygonal and curvilinear areas.
• Second-Degree Coefficient Invariance: In the general second-degree equation $a x^2 + 2h xy + b y^2 + 2g x + 2f y + c = 0$, pure translation changes only the linear terms ($g, f$) and the constant $c$, leaving the second-degree quadratic coefficients $a, h, b$ strictly invariant!

1. Mathematical Formulation of Axis Rotation

Let the Cartesian axes $Ox, Oy$ be rotated counterclockwise through an angle $\theta$ about the fixed origin $O$ to a new coordinate system $OX, OY$. Let a point $P$ have polar coordinates $(r, \phi)$ with respect to the original system, so that $x = r \cos \phi$ and $y = r \sin \phi$. In the rotated system, the distance $r$ remains unchanged, but the angle from the $OX$-axis to $OP$ is $\phi - \theta$. Therefore: $$X = r \cos(\phi - \theta) = r \cos \phi \cos \theta + r \sin \phi \sin \theta = x \cos \theta + y \sin \theta$$ $$Y = r \sin(\phi - \theta) = r \sin \phi \cos \theta - r \cos \phi \sin \theta = -x \sin \theta + y \cos \theta$$ Inverting these linear relations gives the old coordinates $(x, y)$ in terms of the new coordinates $(X, Y)$: $$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} \iff \begin{cases} x = X \cos \theta - Y \sin \theta \\ y = X \sin \theta + Y \cos \theta \end{cases}$$ The transformation matrix: $$\mathbf{R}(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$ is a special orthogonal matrix belonging to the Lie group $\mathrm{SO}(2)$, satisfying $\mathbf{R}^T \mathbf{R} = \mathbf{I}$ and $\det \mathbf{R} = +1$.

2. The Elimination of the Cross-Product Term ($xy$)

Under rotation of axes through an angle $\theta$, the general second-degree form $a x^2 + 2h xy + b y^2$ transforms into $A X^2 + 2H XY + B Y^2$. Expanding and grouping terms: $$A = a \cos^2 \theta + 2h \sin \theta \cos \theta + b \sin^2 \theta$$ $$B = a \sin^2 \theta - 2h \sin \theta \cos \theta + b \cos^2 \theta$$ $$2H = 2(b - a)\sin \theta \cos \theta + 2h(\cos^2 \theta - \sin^2 \theta) = (b - a)\sin 2\theta + 2h \cos 2\theta$$ To eliminate the cross-product term $XY$, we require $H = 0$: $$(b - a)\sin 2\theta + 2h \cos 2\theta = 0 \iff (a - b)\sin 2\theta = 2h \cos 2\theta$$ $$\tan 2\theta = \frac{2h}{a - b} \quad (a \ne b), \qquad \text{or } \theta = \frac{\pi}{4} \quad (a = b)$$ This fundamental relation is the cornerstone of conic canonical reduction!

3. Rotational Invariants of Quadratic Forms

For any rotation of axes, the coefficients of the quadratic form satisfy two algebraic invariants: $$\mathbf{Invariant \; 1:} \quad A + B = a + b \quad (\text{Trace of the matrix})$$ $$\mathbf{Invariant \; 2:} \quad AB - H^2 = ab - h^2 \quad (\text{Determinant of the matrix})$$ These invariants ensure that the geometric nature of the quadratic curve (ellipse, parabola, or hyperbola) is an intrinsic property of the curve, independent of the choice of coordinate axes!