D. Examples

1. Three point curve

Three known coordinate points are

Determine the radius of the circle passing through the points and coordinates of the radius point.

Sketch

Compute the coefficients using Equations I-3 through I-6

Use the coefficients in Equations I-1 and I-2 to get radius point coordinates

Get radius from Equation I-7

Using point A

Check using point B

Checks within rounding.

Answers

2. Tangent line

What is the direction of the line line tangent to the arc shown below? What are the coordinates of the intersection point?

Inverse from point W to the radius point O

Using Equation I-12 compute the angle g between the line to the radius point and tangent line.

Determine the azimuth and length of the tangent line

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Since the tangent line is left of the line to the radius point, to get its azimuth subtract g from Az_WO' Equation, I-16.

Its length comes from Equation I-13

Forward Computation, Equations I-17 and I-18, to compute tangent point coordinates

Answers

3. Three-tangent arc

What is the radius of the arc that is tangent to the three line shown?

Label the geometry

Establsh points G and H using direction-direction intersections (results shown, comps left to user)

Δ_G and Δ_H from the azimuths (comps left to user)

Inverse Computation to obtain distance from point G to point H  (comps left to user): 383.084

Apply Equation I-19 to get the arc radius

Tangent distance for first arc section (comps left to user): 219.721

Radius point coordinates (comps left to user): 1,442.592' N and 1,408.580' E

Answers