Add a property on Arc3d for the PI

[FelixGirard]

Is your feature request related to a problem? Please describe.
Currently, we have some code to compute the PI of an Arc3d:

Describe the solution you'd like

I would like piPoint to just be a property on the Arc3d.

[dassaf4]

A few comments:

• As this is a nontrivial computation, I would rather it not be a property.
• We would probably call it something other than piPoint, as this term is not widely known (outside Civil context).
• We would probably compute with an xy-intersection in local coordinates (yields same result).
• My gut feeling is to return undefined for an arc that is within tight tolerance of a semicircle.

[saeeedtorabi]

I'm not familiar with the context so I dont know why the civil team calls it "pi point" or dont even know the relationship of this point and pi.

This is my understanding:
The "pi point" is the intersection point between tangent lines at the start and end of a partial arc.

Let me explain by an example:
Suppose the arc is a unit circle and we want to find "pi point" for partial arcs starting at (1,0) increasing the angle sweep from 0 to 360. For angle 0, the pi point starts at (1,0) and as we increase the sweep, the pi point moves along the line x = 1 (blue line in below image). The pi point goes to infinity as we get close to sweep pi (180 degrees). If we continues, the pi point jumps to minus infinity and moves up along the blue line until it approaches to (1,0) as we approach to sweep 360.

Another observation was if we connect the pi point and arc center by a line segment, the intersection of the line segment and the arc, would be the middle point of the arc.

Another example with

const origin = Point3d.create();
const unitX = Vector3d.create(1, 0, 0);
const unitY = Vector3d.create(1, 1, 0);
const arc = Arc3d.create(origin, unitX, unitY, AngleSweep.createStartEndDegrees(0, 180));

The 2 blue lines are clearly tangents at start and end. They are parallel but debug says the approach type is CurveCurveApproachType.PerpendicularChord. On the other hand, Ray3d.closestApproachRay3dRay3d is returning the small white circle which is intersection of tangent at start and the white line (not sure where the white line is coming from). The intersection of white line and arc happens at angle pi of the arc so I assume that's why it's called "pi point" but I dont get how the white line is generated.

Until I understand the meaning of the "pi point" I cannot find a better name for it and also dont understand why it can be also calculated by xy-intersection in local coordinates.

Test code:

  function piPoint(arc3d: Arc3d): Point3d {
    const startPoint = arc3d.fractionToPointAndUnitTangent(0.0);
    console.log(startPoint.direction.toJSON());
    const endPoint = arc3d.fractionToPointAndUnitTangent(1.0);
    console.log(endPoint.direction.toJSON());
    const intersection = Ray3d.closestApproachRay3dRay3d(startPoint, endPoint);
    return intersection.detailA.point;
  }
  it.only("PiPoint", () => {
    const ck = new Checker(true, true);
    const allGeometry: GeometryQuery[] = [];

    const origin = Point3d.create();
    const unitX = Vector3d.create(1, 0, 0);
    const unitY = Vector3d.create(1, 1, 0);
    const arc = Arc3d.create(origin, unitX, unitY, AngleSweep.createStartEndDegrees(0, 180));
    GeometryCoreTestIO.captureCloneGeometry(allGeometry, arc);
    const p = piPoint(arc);
    console.log(p.toJSON());
    GeometryCoreTestIO.createAndCaptureXYCircle(allGeometry, p, 0.1);

    GeometryCoreTestIO.saveGeometry(allGeometry, "Arc3d", "PiPoint");
    expect(ck.getNumErrors()).toBe(0);
  });

[a-gagnon]

I've been asked to provide some details.

This is my understanding:
The "pi point" is the intersection point between tangent lines at the start and end of a partial arc.

@saeeedtorabi That's exactly what it is. PI stands for 'Point of Intersection'. In civil design, circular arcs are typically constructed having tangent lines and radius as input. I believe similar work is done inside the function CurveFactory.createFilletsInLineString.

As David already pointed out, it is undefined for semicircles.
It's also worth mentioning that some ways of computing the PI don't do well for arcs with no length (ie. sweep = 0). Those rarely arise in user datasets, but may happen when first laying out civil alignments.

As this is seemingly civil specific, I don't have a strong opinion and will leave it to you guys to determine if this should be in core-geometry or not.

Thx,
Alex

[FelixGirard]

Since we are on the topic, there is also the reverse of CurveFactory.createFilletsInLineString that we are implementing and could be part of core. (parsing point and radii from a filleted line string)

[dassaf4]

Since we are on the topic, there is also the reverse of CurveFactory.createFilletsInLineString that we are implementing and could be part of core. (parsing point and radii from a filleted line string)

Sounds reasonable to provide a filleted line string parser in CurveFactory. @FelixGirard can you start a new Issue for this?

[FelixGirard]

Done!: #8621
Thanks