Skew Lines Distance⚠ unverified

Understanding the Parameters

• x_1 (m) — Length.
• y_1 (m) — Length.
• z_1 (m) — Length.
• d_{1x} (m) — Length.
• d_{1y} (m) — Length.
• d_{1z} (m) — Length.
• x_2 (m) — Length.
• y_2 (m) — Length.
• z_2 (m) — Length.
• d_{2x} (m) — Length.
• d_{2y} (m) — Length.
• d_{2z} (m) — Length.
• Output d (m) — Distance.

How to Calculate

1. Enter Length as x1 (default 0 m). Use the unit menu when you need a different unit.
2. Enter Length as y1 (default 0 m). Use the unit menu when you need a different unit.
3. Enter Length as z1 (default 0 m). Use the unit menu when you need a different unit.
4. Enter Length as dx1 (default 1 m). Use the unit menu when you need a different unit.
5. Enter Length as dy1 (default 0 m). Use the unit menu when you need a different unit.
6. Enter Length as dz1 (default 0 m). Use the unit menu when you need a different unit.
7. Enter Length as x2 (default 0 m). Use the unit menu when you need a different unit.
8. Enter Length as y2 (default 1 m). Use the unit menu when you need a different unit.
9. Enter Length as z2 (default 1 m). Use the unit menu when you need a different unit.
10. Enter Length as dx2 (default 0 m). Use the unit menu when you need a different unit.
11. Enter Length as dy2 (default 1 m). Use the unit menu when you need a different unit.
12. Enter Length as dz2 (default 0 m). Use the unit menu when you need a different unit.
13. Click Calculate. The card evaluates $d=|(P_2-P_1)\cdot(d_1\times d_2)|/|d_1\times d_2|$ 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.

• 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