Tangents to circle from point and intersection line/circle

Grasshopper
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1

I am trying to create an animation showing a collision avoidance system. To do it, I need to find and display the evasion courses at every point in the animation. For that reason, I would like to have, if possible, the logic encapsulated in a single C# component.

The procedure is straightforward geometrically. It is shown in the following diagram:

diagram
diagram2267×1275 193 KB

In essence, you need to find the two tangents from the target to a circle that represents the minimum desired distance centered on our ship. Then determine the intersection of these tangent lines with a circle that has as a radius our ship’s speed and is placed at the end point of the target ship velocity vector.

I have not been able to find suitable RhinoCommon functions, and my analytic geometry skills are not what they used to be.

Any tips will be most welcomed.

2

You can find the tangent points by constructing a circle with diameter (ship position, target position) and intersecting it with your minimum distance circle.

Circle intersection algorithm here

3

Here’s how I would do it in Python using the rhinoscriptsyntax methods for curve intersection.

import Rhino
import rhinoscriptsyntax as rs

target_start = Rhino.Geometry.Point3d(T0)
target_end = Rhino.Geometry.Point3d(T1)
self_start = Rhino.Geometry.Point3d(S0)
self_end = Rhino.Geometry.Point3d(S1)
min_distance = float(D)

def curve_intersection(curve1, curve2):
    results = rs.CurveCurveIntersection(curve1, curve2)
    points = []
    for result in results:
        event, location = result[:2]
        if event == 1: points.append(location)
    return points

circle_distance = rs.AddCircle(self_start, min_distance)
midpoint = (self_start + target_start)/2
separation = self_start.DistanceTo(target_start)
circle_separation = rs.AddCircle(midpoint, separation/2)
tangent_points = curve_intersection(circle_distance, circle_separation)
tangent_lines = [rs.AddLine(target_start, Rhino.Geometry.Point3d(pt)) for pt in tangent_points]
self_speed = self_start.DistanceTo(self_end)
circle_speed = rs.AddCircle(target_end, self_speed)
intersect_points = sum([curve_intersection(circle_speed, line) for line in tangent_lines], start=[])
evasion_points = sorted(intersect_points, key=lambda pt: pt.DistanceTo(self_start))[:2]
evasion_points = [self_start + target_end - pt for pt in evasion_points]

EP = evasion_points
4

rhinocommon has a lot of intersection methods

https://developer.rhino3d.com/api/rhinocommon/rhino.geometry.intersect.intersection

you might use curve-curve intersection or Circle Circle
https://developer.rhino3d.com/api/rhinocommon/rhino.geometry.intersect.intersection/circlecircle

there is also
https://developer.rhino3d.com/api/rhinocommon/rhino.geometry.intersect.intersection/linecircle

note the speciality of line parameters
t Type: double
Parameter to evaluate line segment at. Line parameters are normalized parameters.
https://developer.rhino3d.com/api/rhinocommon/rhino.geometry.line/pointat#(double)

the rs. … commands that are used by @Measure will (and need) RhinoObjects being added to the document. This is not nice in a Gh-component.
if you really want to use this approach, make sure to delete the objects after the solution is computed.

for the initial tangents there is:
https://developer.rhino3d.com/api/rhinocommon/rhino.geometry.curve/getlocaltangentpoint

don’t forget at the beginning of your scipt to check that the setup is planar.

hope above or Meassure’s post help as a starting point - feel free to post some code or gh-definition to get further feedback.

happy coding - kind regards -tom

5

@Measure
Thank you very much for your code. It works beautifully!

Now, all I need to do is to translate it to C# and incorporate it to my solution.

6

Tom,

Thank you very much for your information. Actually, I already solved myself the first part of the problem, get the tangents to a circle through a point, using simple trigonometry.

In case someone is interested, here is the solution to find the tangents to a circle from an external point:

        // Tangents
        Double Theta = Math.Asin(minDist / dist);
        Double tanDist = Math.Sqrt(dist * dist - minDist * minDist);
        Vector3d v1 = ownOrig - tgtOrig;
        v1.Unitize();
        v1.Rotate(Theta, -Vector3d.ZAxis);
        Line tangent1 = new Line(tgtOrig, tgtOrig + v1 * tanDist);

        Vector3d v2 = ownOrig - tgtOrig;
        v2.Unitize();
        v2.Rotate(-Theta, -Vector3d.ZAxis);
        Line tangent2 = new Line(tgtOrig, tgtOrig + v2 * tanDist);

It is really straight forward and gives very good results. Actually, this was my first intuition, but I made a mistake in my first implementation, and discarded the method before posting my question.

Finally, the implementation of the intersection between the circle and the two tangent lines is similarly easy:

        Circle velCircle = new Circle(tgtVel, ownSpeed / 15);
        circles.Add(velCircle);
        Double t1, t2;
        Point3d p1, p2, p3, p4;
        Rhino.Geometry.Intersect.Intersection.LineCircle(tangent1, velCircle, 
            out t1, out p1, out t2, out p2);

        Rhino.Geometry.Intersect.Intersection.LineCircle(tangent2, velCircle, 
            out t1, out p3, out t2, out p4);