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Index/Geometry/GMY-05

GMY-05

3-D distance

Euclidean.

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Geometry3D distance

Governing equation

d=sqrtDeltax2+Deltay2+Deltaz2d=\\sqrt{\\Delta x^2+\\Delta y^2+\\Delta z^2}

Δx∈[−20,20]\Delta x \in \left[-20, 20\right]

where

\Delta x
Δx (m)
\Delta y
Δy (m)
\Delta z
Δz (m)
d
d (m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

d=sqrtDeltax2+Deltay2+Deltaz2d=\\sqrt{\\Delta x^2+\\Delta y^2+\\Delta z^2}

Δx∈[−20,20]\Delta x \in \left[-20, 20\right]

Δx=−20\Delta x = -20d=23.66d = 23.66

②

Domain of the variable — blue

Δxmin⁡=−20,Δxmax⁡=20\Delta x_{\min} = -20,\quad \Delta x_{\max} = 20

Δx∈[−20,20]\Delta x \in \left[-20, 20\right]

Δx=−20\Delta x = -20Δx=−20\Delta x = -20

③

Response curve — green

d=sqrtDeltax2+Deltay2+Deltaz2d=\\sqrt{\\Delta x^2+\\Delta y^2+\\Delta z^2}

Δx∈[−20,20]\Delta x \in \left[-20, 20\right]

Δx=−20\Delta x = -20d=23.66d = 23.66

④

Rate of change — amber

dddΔx\frac{\mathrm{d}d}{\mathrm{d}\Delta x}

Δx∈[−20,20]\Delta x \in \left[-20, 20\right]

Δx=−19.75\Delta x = -19.75d′=−0.842d^{\prime} = -0.842

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-05 — 3-D distance) is the form associated with 3D distance. Working symbols: Δx\Delta x, Δy\Delta y, Δz\Delta z →\rightarrow dd. Euclidean.

Purpose

Purpose: compute dd from Δx\Delta x, Δy\Delta y, Δz\Delta z in Geometry via d=sqrtDeltax2+Deltay2+Deltaz2d=\\sqrt{\\Delta x^2+\\Delta y^2+\\Delta z^2} Euclidean. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given Δx=3.000 m\Delta x = 3.000\,\mathrm{m}, Δy=4.000 m\Delta y = 4.000\,\mathrm{m}, Δz=12.000 m\Delta z = 12.000\,\mathrm{m}, the governing relation d=sqrtDeltax2+Deltay2+Deltaz2d=\\sqrt{\\Delta x^2+\\Delta y^2+\\Delta z^2} yields d=13.0000 md = 13.0000\,\mathrm{m}. Euclidean. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • d d13.0000 m
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Narration of this film

Euclidean.

Euclidean.

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