find tangent line of two adjacent circle - math

Question

those are 2 example cases of what I need to solve, it is just finding the coordinate of D, given position of A, and the direction vector of red and green line
red/green line vector (or direction) is known
point A is an intersection between the red line and red circle tangent point
point B is the center of the red circle with radius = R (known)
point C is an intersection between the green line and the green circle tangent point
point D is unknown and this one that needs to be calculated
point D will always located in green circle (radius of 2R from point B)
both red and green line has the same radius of R
V is the angle of the red line relative to north up
W is the angle of the green line relative to north up
the distance between point B and D is always 2R since the circle adjacent (touching each other)
much help and hint appreciated, preferred in some code instead of math equation

Answer 1

Having coordinates A,B,C, we can write two vector equations using scalar (dot) product:
AC.dot.DC = 0
DB.dot.DB = 4*R^2
The first one refers to perpendicularity between tangent to circle and radius to tangency point, the second one - just squared distance between circle centers.
In coordinates:
(cx-ax)*(cx-dx) + (cy-ay)*(cy-dy) = 0
(bx-dx)*(bx-dx) + (by-dy)*(by-dy) = 4*R^2
Solve this system for unknown dx, dy - two solutions in general case.
If A and C are not known, as #Mike 'Pomax' Kamermans noticed:
Let
cr = sin(v) sr = cos(v)
cg = sin(w) sg = cos(w)
So
ax = bx + R * cr
ay = by + R * sr
and
dx = cx - R * cg
dy = cy + R * sg
Substituting expressions into the system above we have:
(dx+R*cg-bx-R*cr)*cg - (dy-R*sg-by-R*sr)*sg = 0
(bx-dx)*(bx-dx) + (by-dy)*(by-dy) = 4*R^2
Again - solve system for unknowns dx, dy

Answer 2

As a hint: draw it out some more:
We can construct D by constructing the line segment AG, for which we know both the angle and length, because AC⟂AG, and the segment has length R.
We can then construct a line perpendicular to AG, through G, which gives us a chord on the blue circle, one endpoint of which is D. We know the distance from B to GD (because we know trigonometry) and we know that the distance BD is 2R (because that's a given). Pythagoras then trivially gives us D.

Answer 3

You know where A is and the angle θ it makes from vertical.
So specify the line though C called line(C) above and the offset the line by R in order to get line(D) above that goes through point D.
In C# code this is
Line line_C = Line.ThroughPointAtAngle(A, theta);
Line line_D = line_C.Offset(radius);
Now find the intersection of this line to the greater circle
Circle circle = new Circle(B, 2 * radius);
if (circle.Intersect(line_D, out Point D, alternate: false))
{
Console.WriteLine(D);
float d_BD = B.DistanceTo(D);
Console.WriteLine(d_BD);
}
else
{
Console.WriteLine("Does not intersect.");
}
This produces point D either above line(C) or below line(C) depending on the bool argument alternate.
The code example below produces the following output:
D=Point(-0.4846499,-1.94039)
|BD|=2
The source code is
Program.cs
using static Float;
static class Program
{
static void Main(string[] args)
{
float radius = 1;
Point A = new Point(-radius, 0);
Point B = new Point(0, 0);
float theta = deg(15);
Line line_C = Line.ThroughPointAtAngle(A, theta);
Line line_D = line_C.Offset(radius);
Circle circle = new Circle(B, 2 * radius);
if (circle.Intersect(line_D, out Point D, alternate: false))
{
Console.WriteLine($"D={D}");
float d_BD = B.DistanceTo(D);
Console.WriteLine($"|BD|={d_BD}");
}
else
{
Console.WriteLine("Does not intersect.");
}
}
}
Point.cs
Describes a point in cartesian space using two coordinates (x,y)
using static Float;
public readonly struct Point
{
readonly (float x, float y) data;
public Point(float x, float y)
{
this.data = (x, y);
}
public static Point Origin { get; } = new Point(0, 0);
public static Point FromTwoLines(Line line1, Line line2)
{
float x = line1.B * line2.C - line1.C * line2.B;
float y = line1.C * line2.A - line1.A * line2.C;
float w = line1.A * line2.B - line1.B * line2.A;
return new Point(x / w, y / w);
}
public float X => data.x;
public float Y => data.y;
public float SumSquares => data.x * data.x + data.y * data.y;
#region Algebra
public static Point Negate(Point a)
=> new Point(
-a.data.x,
-a.data.y);
public static Point Scale(float factor, Point a)
=> new Point(
factor * a.data.x,
factor * a.data.y);
public static Point Add(Point a, Point b)
=> new Point(
a.data.x + b.data.x,
a.data.y + b.data.y);
public static Point Subtract(Point a, Point b)
=> new Point(
a.data.x - b.data.x,
a.data.y - b.data.y);
public static float Dot(Point point, Line line)
=> line.A * point.data.x + line.B * point.data.y + line.C;
public static Point operator +(Point a, Point b) => Add(a, b);
public static Point operator -(Point a) => Negate(a);
public static Point operator -(Point a, Point b) => Subtract(a, b);
public static Point operator *(float f, Point a) => Scale(f, a);
public static Point operator *(Point a, float f) => Scale(f, a);
public static Point operator /(Point a, float d) => Scale(1 / d, a);
#endregion
#region Geometry
public Point Offset(float dx, float dy)
=> new Point(data.x + dx, data.y + dy);
public Point Offset(Vector2 delta) => Offset(delta.X, delta.Y);
public float DistanceTo(Point point)
=> sqrt(sqr(data.x - point.data.x) + sqr(data.y - point.data.y));
#endregion
#region Formatting
public string ToString(string formatting, IFormatProvider provider)
{
return $"Point({data.x.ToString(formatting, provider)},{data.y.ToString(formatting, provider)})";
}
public string ToString(string formatting)
=> ToString(formatting, null);
public override string ToString()
=> ToString("g");
#endregion
}
Line.cs
Describes a line in cartesian space using the coefficients (a,b,c) such that the equation of the line is a x + b y + c = 0
using static Float;
public readonly struct Line
{
readonly (float a, float b, float c) data;
public Line(float a, float b, float c) : this()
{
data = (a, b, c);
}
public static Line AlongX { get; } = new Line(0, 1, 0);
public static Line AlongY { get; } = new Line(-1, 0, 0);
public static Line ThroughPointAtAngle(Point point, float angle)
{
return new Line(cos(angle), -sin(angle), point.Y * sin(angle) - point.X * cos(angle));
}
public static Line ThroughTwoPoints(Point point1, Point point2)
=> new Line(
point1.Y - point2.Y,
point2.X - point1.X,
point1.X * point2.Y - point1.Y * point2.X);
public float A => data.a;
public float B => data.b;
public float C => data.c;
#region Algebra
public static float Dot(Line line, Point point)
=> line.data.a * point.X + line.data.b * point.Y + line.data.c;
#endregion
#region Geometry
public Line ParallelThrough(Point point)
{
return new Line(data.a, data.b, -data.a * point.X - data.b * point.Y);
}
public Line PerpendicularThrough(Point point)
{
return new Line(data.b, -data.a, -data.b * point.X + data.a * point.Y);
}
public Line Offset(float amount)
=> new Line(data.a, data.b, data.c - amount * sqrt(sqr(data.a) + sqr(data.b)));
public Line Offset(float dx, float dy)
=> new Line(data.a, data.b, data.c + data.a * dx + data.b * dy);
public Line Offset(Vector2 delta) => Offset(delta.X, delta.Y);
public float DistanceTo(Point point)
=> Dot(this, point) / (data.a * data.a + data.b * data.b);
#endregion
#region Formatting
public string ToString(string formatting, IFormatProvider provider)
{
return $"Line({data.a.ToString(formatting, provider)}x+{data.b.ToString(formatting, provider)}y+{data.c.ToString(formatting, provider)}=0)";
}
public string ToString(string formatting)
=> ToString(formatting, null);
public override string ToString()
=> ToString("g");
#endregion
}
Circle.cs
Describes a circle using the center and radius.
using static Float;
public readonly struct Circle
{
readonly (Point center, float radius) data;
public Circle(Point center, float radius)
{
this.data = (center, radius);
}
public static Circle FromTwoPoints(Point point1, Point point2)
{
float radius = point1.DistanceTo(point2) / 2;
Point center = (point1 + point2) / 2;
return new Circle(center, radius);
}
public static Circle FromThreePoints(Point point1, Point point2, Point point3)
{
float k_1 = point1.SumSquares / 2;
float k_2 = point2.SumSquares / 2;
float k_3 = point3.SumSquares / 2;
float dx_12 = point2.X - point1.X;
float dy_12 = point2.Y - point1.Y;
float dx_23 = point3.X - point2.X;
float dy_23 = point3.Y - point2.Y;
float det = dx_12 * dy_23 - dx_23 * dy_12;
Point center = new Point(
(dy_12 * (k_2 - k_3) + dy_23 * (k_2 - k_1)) / det,
(dx_12 * (k_3 - k_2) + dx_23 * (k_1 - k_2)) / det);
float radius = center.DistanceTo(point1);
return new Circle(center, radius);
}
public Point Center => data.center;
public float Radius => data.radius;
#region Geometry
public float DistanceTo(Point point)
=> data.center.DistanceTo(point) - data.radius;
public float DistanceTo(Line line)
{
float d = line.DistanceTo(Center);
if (d > 0)
{
return d - data.radius;
}
else
{
return d + data.radius;
}
}
public bool Intersect(Line line, out Point point, bool alternate = false)
{
line = line.Offset(-Center.X, -Center.Y);
int sign = alternate ? -1 : 1;
float discr = sqr(line.A * data.radius) + sqr(line.B * data.radius) - sqr(line.C);
if (discr >= 0)
{
float d = sign * sqrt(discr);
float ab = line.A * line.A + line.B * line.B;
point = new Point((line.B * d - line.A * line.C) / ab, -(line.A * d + line.B * line.C) / ab);
point += Center;
return true;
}
else
{
float ab = line.A * line.A + line.B * line.B;
point = new Point((-line.A * line.C) / ab, -(+line.B * line.C) / ab);
point += Center;
return false;
}
}
#endregion
#region Formatting
public string ToString(string formatting, IFormatProvider provider)
{
return $"Circle({data.center.ToString(formatting, provider)},{data.radius.ToString(formatting, provider)})";
}
public string ToString(string formatting)
=> ToString(formatting, null);
public override string ToString()
=> ToString("g");
#endregion
}
Float.cs
Helper functions dealing with float math which is lacking from System.Math.
public static class Float
{
/// <summary>
/// A factor of π.
/// </summary>
/// <param name="x">The factor.</param>
public static float pi(float x) => (float)(Math.PI * x);
/// <summary>
/// Degree to Radian conversion
/// </summary>
/// <param name="x">The angle in degrees.</param>
/// <returns>Angle in radians</returns>
public static float deg(float x) => pi(x) / 180;
/// <summary>
/// Radian to Degree conversion
/// </summary>
/// <param name="x">The angle in radians.</param>
/// <returns>Angle in degrees</returns>
public static float rad(float x) => x * 180 / pi(1);
public static float sqr(float x) => x * x;
public static float sqrt(float x) => (float)Math.Sqrt(x);
public static float sin(float x) => (float)Math.Sin(x);
public static float cos(float x) => (float)Math.Cos(x);
}

Answer 4

thx for all the answer (will upvote them), I did the work on it myself, I will share my result :
and here is length of d, to solve the equation :