Point To Line Distance⚠ unverified
Geometry / Analytic · Distance from a point to ax+by+c=0
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| a | a | — | 1.0 | Coefficient a (1/m) |
| b | b | — | -1.0 | Coefficient b (1/m) |
| c | c | — | 0.0 | Coefficient c |
| x0 | x0 | m | 0 | Point x |
| y0 | y0 | m | 1 | Point y |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| d | d | m | Distance |
The science & history
Understanding the Parameters
- a — Coefficient a (1/m).
- b — Coefficient b (1/m).
- c — Coefficient c.
- x_0 (m) — Point x.
- y_0 (m) — Point y.
- Output d (m) — Distance.
How to Calculate
- Enter Coefficient a (1/m) as
a(default 1.0 dimensionless). Use the unit menu when you need a different unit. - Enter Coefficient b (1/m) as
b(default -1.0 dimensionless). Use the unit menu when you need a different unit. - Enter Coefficient c as
c(default 0.0 dimensionless). Use the unit menu when you need a different unit. - Enter Point x as
x0(default 0 m). Use the unit menu when you need a different unit. - Enter Point y as
y0(default 1 m). Use the unit menu when you need a different unit. - Click Calculate. The card evaluates $d=|ax_0+by_0+c|/\sqrt{a^2+b^2}$ 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.
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Euler, Lagrange. Rotation of axes to eliminate $xy$ terms; classification of the general conic.
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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, Circle Radius From General Form, Parabola Latus Rectum, Ellipse Eccentricity, Ellipse Linear Eccentricity, Hyperbola Eccentricity