Cartesian X From Polar⚠ unverified
Geometry / Analytic · Cartesian x from polar coordinates
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| r | r | m | 2 | Radius |
| theta | θ | degree | 60 | Angle |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| x | x | m | x coordinate |
The science & history
Understanding the Parameters
- r (m) — Radius.
- θ (degree) — Angle.
- Output x (m) — x coordinate.
How to Calculate
- Enter Radius as
r(default 2 m). Use the unit menu when you need a different unit. - Enter Angle as
theta(default 60 degree). Use the unit menu when you need a different unit. - Click Calculate. The card evaluates $x=r\cos\theta$ and shows the result in the declared output unit; Result in (when present) converts that number.
The Science: A Rigorous Derivation (Approaching a Proof)
Analytic geometry replaces figures with equations. A point is a tuple of real numbers; the Euclidean distance is the $2$-norm of their difference. A line $ax+by+c=0$ is a level set of a linear function, so the distance from a point to the line is that function's value divided by the norm of its gradient. Conics are the quadratic curves; completing the square on $x^2+y^2+Dx+Ey+F=0$ yields a circle; $e = c/a$ classifies ellipse ($e<1$), parabola ($e=1$) and hyperbola ($e>1$). Polar coordinates $(r,\theta)$ are the Euclidean plane written in rotationally natural coordinates.
Dimensional check. The declared output unit is m; ToolBox evaluates the
formula in SI (radians internally for every trigonometric call) and python-calc converts to
the unit on the card.
History and Development
- Descartes, La Géométrie (1637), and Fermat. Coordinates as a method; the name “Cartesian.”
-
Apollonius of Perga (3rd century BCE). Conics — ellipse, parabola, hyperbola — centuries before coordinates, in purely geometric language.
-
Euler, Lagrange. Rotation of axes to eliminate $xy$ terms; classification of the general conic.
-
19th century. Vector geometry (Gibbs, Heaviside) rewrites point-to-plane and skew-line distance as $|(\mathbf{P}_2-\mathbf{P}_1)\cdot(\mathbf{d}_1\times\mathbf{d}_2)|/|\mathbf{d}_1\times\mathbf{d}_2|$.
Related Concepts: Distance 3D, Slope, Angle Between Two Lines, Point To Line Distance, Circle Radius From General Form, Parabola Latus Rectum, Ellipse Eccentricity, Ellipse Linear Eccentricity