Hi, I’m trying to write a module that draws an tangential arc between two circles (based on https://www.frassek.org/2d-mathe/kreise-verbinden/).

In principle it works but there are situations where it does not.

The sample code below echos a nan when P2 has been calculated. My assumption is that the issue is based on a loose of precision after calculation of point M.

Can anybody give me a hint how to solve this. Thank you in advance.

$fa = 0.01;

$fs = 0.01;

C1 = [0, 0];

r1 = 20;

C2 = [50, 50];

r2 = 2.5;

R = 90;

test(C1, r1, C2, r2, R);

module test(C1, r1, C2, r2, R) {

 R1 = R - r1;

 R2 = R - r2;

 M = getIntersectionPointsOfTwoCircles2D(C1, R1, C2, R2)[0];

 P1 = getIntersectionPointsOfTwoCircles2D(C1, r1, M, R)[0];

 P2 = getIntersectionPointsOfTwoCircles2D(C2, r2, M, R)[0];

 echo(M, P1, P2);

 translate(C1) circle(r=r1);

 translate(C2) circle(r=r2);

 color("Cyan") polygon(points = [P1, P2, M]);

 color("Blue") translate([0, 0, -2]) circleWithCenter2D(r=R, C=M);

 polygonWithRadiusBetweenTwoPoints2D(P1, P2, M, r=R);

}

module polygonWithRadiusBetweenTwoPoints2D (P1, P2, C, r=1) {

 startAngle = getAngleBetweenTwoPoints2D(C, P1);

 endAngle = startAngle + getAngleBetweenThreePoints2D(P1, P2, C);

 angles=[ for (i = [startAngle:endAngle-1]) i ];

 coords=[ for (j=concat(angles, [endAngle])) [r*cos(j), r*sin(j)] ];

 translate(C) polygon(coords);

}

module circleWithCenter2D(r=1, d, C) {

 translate(C) circle(r=r);

}

module rotateAboutPoint(a, v, pt) {

 translate(pt)

     rotate(a,v)

         translate(-pt)

             children();

}

// getDistanceBetweenTwoPoints2D

// returns the distance between 2 points

// P1: point 1

// P2: point 2

function getDistanceBetweenTwoPoints2D(P1, P2) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

     sqrt((x1 - x2)^2 + (y1 - y2)^2);

// getAngleBetweenThreePoints2D

// returns the angle between 3 points

// P1: point 1

// P2: point 2

// C: center point at angle

function getAngleBetweenThreePoints2D(P1, P2, C) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1], xC=C[0], yC=C[1])

     atan2(y2 - yC, x2 - xC) - atan2(y1 - yC, x1 - xC);

// getAngleBetweenTwoPoints2D

// returns the angle between 2 points

// P1: point 1

// P2: point 2

function getAngleBetweenTwoPoints2D(P1, P2) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

     atan2(y2 - y1, x2 - x1);

// getIntersectionPointsOfTwoCircles2D

// returns 2 possible intersection points of 2 circles

// C1: center of circle 1

// r1: radius of circle 1

// C2: center of circle 2

// r2: radius of circle 2

function getIntersectionPointsOfTwoCircles2D(C1, r1, C2, r2) =

 let(

     x1=C1[0], y1=C1[1], x2=C2[0], y2=C2[1],

     d=sqrt((x2-x1)^2 + (y2-y1)^2),

     a=(r1^2-r2^2+d^2)/(2*d),

     h=sqrt(r1^2-a^2),

     x3=x1+a*(x2-x1)/d,

     y3=y1+a*(y2-y1)/d,

     x4=x3+h*(y2-y1)/d,

     y4=y3-h*(x2-x1)/d,

     x5=x3-h*(y2-y1)/d,

     y5=y3+h*(x2-x1)/d

 ) [[x4, y4], [x5, y5]];

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Hi, I’m trying to write a module that draws an tangential arc between two circles (based on https://www.frassek.org/2d-mathe/kreise-verbinden/).

In principle it works but there are situations where it does not.

The sample code below echos a nan when P2 has been calculated. My assumption is that the issue is based on a loose of precision after calculation of point M.

Can anybody give me a hint how to solve this. Thank you in advance.

$fa = 0.01;

$fs = 0.01;

C1 = [0, 0];

r1 = 20;

C2 = [50, 50];

r2 = 2.5;

R = 90;

test(C1, r1, C2, r2, R);

module test(C1, r1, C2, r2, R) {

 R1 = R - r1;

 R2 = R - r2;

 M = getIntersectionPointsOfTwoCircles2D(C1, R1, C2, R2)[0];

 P1 = getIntersectionPointsOfTwoCircles2D(C1, r1, M, R)[0];

 P2 = getIntersectionPointsOfTwoCircles2D(C2, r2, M, R)[0];

 echo(M, P1, P2);

 translate(C1) circle(r=r1);

 translate(C2) circle(r=r2);

 color("Cyan") polygon(points = [P1, P2, M]);

 color("Blue") translate([0, 0, -2]) circleWithCenter2D(r=R, C=M);

 polygonWithRadiusBetweenTwoPoints2D(P1, P2, M, r=R);

}

module polygonWithRadiusBetweenTwoPoints2D (P1, P2, C, r=1) {

 startAngle = getAngleBetweenTwoPoints2D(C, P1);

 endAngle = startAngle + getAngleBetweenThreePoints2D(P1, P2, C);

 angles=[ for (i = [startAngle:endAngle-1]) i ];

 coords=[ for (j=concat(angles, [endAngle])) [r*cos(j), r*sin(j)] 

];

 translate(C) polygon(coords);

}

module circleWithCenter2D(r=1, d, C) {

 translate(C) circle(r=r);

}

module rotateAboutPoint(a, v, pt) {

 translate(pt)

     rotate(a,v)

         translate(-pt)

             children();

}

// getDistanceBetweenTwoPoints2D

// returns the distance between 2 points

// P1: point 1

// P2: point 2

function getDistanceBetweenTwoPoints2D(P1, P2) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

     sqrt((x1 - x2)^2 + (y1 - y2)^2);

// getAngleBetweenThreePoints2D

// returns the angle between 3 points

// P1: point 1

// P2: point 2

// C: center point at angle

function getAngleBetweenThreePoints2D(P1, P2, C) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1], xC=C[0], yC=C[1])

     atan2(y2 - yC, x2 - xC) - atan2(y1 - yC, x1 - xC);

// getAngleBetweenTwoPoints2D

// returns the angle between 2 points

// P1: point 1

// P2: point 2

function getAngleBetweenTwoPoints2D(P1, P2) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

     atan2(y2 - y1, x2 - x1);

// getIntersectionPointsOfTwoCircles2D

// returns 2 possible intersection points of 2 circles

// C1: center of circle 1

// r1: radius of circle 1

// C2: center of circle 2

// r2: radius of circle 2

function getIntersectionPointsOfTwoCircles2D(C1, r1, C2, r2) =

 let(

     x1=C1[0], y1=C1[1], x2=C2[0], y2=C2[1],

     d=sqrt((x2-x1)^2 + (y2-y1)^2),

     a=(r1^2-r2^2+d^2)/(2*d),

     h=sqrt(r1^2-a^2),

     x3=x1+a*(x2-x1)/d,

     y3=y1+a*(y2-y1)/d,

     x4=x3+h*(y2-y1)/d,

     y4=y3-h*(x2-x1)/d,

     x5=x3-h*(y2-y1)/d,

     y5=y3+h*(x2-x1)/d

 ) [[x4, y4], [x5, y5]];

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Hi, I’m trying to write a module that draws an tangential arc between two circles (based on https://www.frassek.org/2d-mathe/kreise-verbinden/).

In principle it works but there are situations where it does not.

The sample code below echos a nan when P2 has been calculated. My assumption is that the issue is based on a loose of precision after calculation of point M.

Can anybody give me a hint how to solve this. Thank you in advance.

$fa = 0.01;

$fs = 0.01;

C1 = [0, 0];

r1 = 20;

C2 = [50, 50];

r2 = 2.5;

R = 90;

test(C1, r1, C2, r2, R);

module test(C1, r1, C2, r2, R) {

 R1 = R - r1;

 R2 = R - r2;

 M = getIntersectionPointsOfTwoCircles2D(C1, R1, C2, R2)[0];

 P1 = getIntersectionPointsOfTwoCircles2D(C1, r1, M, R)[0];

 P2 = getIntersectionPointsOfTwoCircles2D(C2, r2, M, R)[0];

 echo(M, P1, P2);

 translate(C1) circle(r=r1);

 translate(C2) circle(r=r2);

 color("Cyan") polygon(points = [P1, P2, M]);

 color("Blue") translate([0, 0, -2]) circleWithCenter2D(r=R, C=M);

 polygonWithRadiusBetweenTwoPoints2D(P1, P2, M, r=R);

}

module polygonWithRadiusBetweenTwoPoints2D (P1, P2, C, r=1) {

 startAngle = getAngleBetweenTwoPoints2D(C, P1);

 endAngle = startAngle + getAngleBetweenThreePoints2D(P1, P2, C);

 angles=[ for (i = [startAngle:endAngle-1]) i ];

 coords=[ for (j=concat(angles, [endAngle])) [r*cos(j), r*sin(j)] ];

 translate(C) polygon(coords);

}

module circleWithCenter2D(r=1, d, C) {

 translate(C) circle(r=r);

}

module rotateAboutPoint(a, v, pt) {

 translate(pt)

     rotate(a,v)

         translate(-pt)

             children();

}

// getDistanceBetweenTwoPoints2D

// returns the distance between 2 points

// P1: point 1

// P2: point 2

function getDistanceBetweenTwoPoints2D(P1, P2) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

     sqrt((x1 - x2)^2 + (y1 - y2)^2);

// getAngleBetweenThreePoints2D

// returns the angle between 3 points

// P1: point 1

// P2: point 2

// C: center point at angle

function getAngleBetweenThreePoints2D(P1, P2, C) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1], xC=C[0], yC=C[1])

     atan2(y2 - yC, x2 - xC) - atan2(y1 - yC, x1 - xC);

// getAngleBetweenTwoPoints2D

// returns the angle between 2 points

// P1: point 1

// P2: point 2

function getAngleBetweenTwoPoints2D(P1, P2) =

 let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

     atan2(y2 - y1, x2 - x1);

// getIntersectionPointsOfTwoCircles2D

// returns 2 possible intersection points of 2 circles

// C1: center of circle 1

// r1: radius of circle 1

// C2: center of circle 2

// r2: radius of circle 2

function getIntersectionPointsOfTwoCircles2D(C1, r1, C2, r2) =

 let(

     x1=C1[0], y1=C1[1], x2=C2[0], y2=C2[1],

     d=sqrt((x2-x1)^2 + (y2-y1)^2),

     a=(r1^2-r2^2+d^2)/(2*d),

     h=sqrt(r1^2-a^2),

     x3=x1+a*(x2-x1)/d,

     y3=y1+a*(y2-y1)/d,

     x4=x3+h*(y2-y1)/d,

     y4=y3-h*(x2-x1)/d,

     x5=x3-h*(y2-y1)/d,

     y5=y3+h*(x2-x1)/d

 ) [[x4, y4], [x5, y5]];

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Yes, P1 and P2 have only one intersection point. Function getIntersectionPointsOfTwoCircles2D should return the same point twice.
The circle should intersect and not be inside as the mathematical formulas are correct.
If I take C2 = [50, 0]; it works

-----Ursprüngliche Nachricht-----
Von: Adrian Mariano avm4@cornell.edu
Gesendet: Freitag, 7. Januar 2022 14:21
An: OpenSCAD general discussion discuss@lists.openscad.org
Betreff: [OpenSCAD] Re: Tangential arc between two circles

Your two circles are tangent, with one of them inside the other one.
There is only one intersection point, and the notion of the "tangent arcs" makes no sense in this case.

On Fri, Jan 7, 2022 at 7:31 AM Joachim Röttinger joe@joecool.de wrote:

Hi, I’m trying to write a module that draws an tangential arc between two circles (based on https://www.frassek.org/2d-mathe/kreise-verbinden/).

In principle it works but there are situations where it does not.

The sample code below echos a nan when P2 has been calculated. My assumption is that the issue is based on a loose of precision after calculation of point M.

Can anybody give me a hint how to solve this. Thank you in advance.

$fa = 0.01;

$fs = 0.01;

C1 = [0, 0];

r1 = 20;

C2 = [50, 50];

r2 = 2.5;

R = 90;

test(C1, r1, C2, r2, R);

module test(C1, r1, C2, r2, R) {

R1 = R - r1;

R2 = R - r2;

M = getIntersectionPointsOfTwoCircles2D(C1, R1, C2, R2)[0];

P1 = getIntersectionPointsOfTwoCircles2D(C1, r1, M, R)[0];

P2 = getIntersectionPointsOfTwoCircles2D(C2, r2, M, R)[0];

echo(M, P1, P2);

translate(C1) circle(r=r1);

translate(C2) circle(r=r2);

color("Cyan") polygon(points = [P1, P2, M]);

color("Blue") translate([0, 0, -2]) circleWithCenter2D(r=R, C=M);

polygonWithRadiusBetweenTwoPoints2D(P1, P2, M, r=R);

}

module polygonWithRadiusBetweenTwoPoints2D (P1, P2, C, r=1) {

startAngle = getAngleBetweenTwoPoints2D(C, P1);

endAngle = startAngle + getAngleBetweenThreePoints2D(P1, P2, C);

angles=[ for (i = [startAngle:endAngle-1]) i ];

coords=[ for (j=concat(angles, [endAngle])) [r*cos(j), r*sin(j)]

];

translate(C) polygon(coords);

}

module circleWithCenter2D(r=1, d, C) {

translate(C) circle(r=r);

}

module rotateAboutPoint(a, v, pt) {

translate(pt)

    rotate(a,v)

        translate(-pt)

            children();

}

// getDistanceBetweenTwoPoints2D

// returns the distance between 2 points

// P1: point 1

// P2: point 2

function getDistanceBetweenTwoPoints2D(P1, P2) =

let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

    sqrt((x1 - x2)^2 + (y1 - y2)^2);

// getAngleBetweenThreePoints2D

// returns the angle between 3 points

// P1: point 1

// P2: point 2

// C: center point at angle

function getAngleBetweenThreePoints2D(P1, P2, C) =

let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1], xC=C[0], yC=C[1])

    atan2(y2 - yC, x2 - xC) - atan2(y1 - yC, x1 - xC);

// getAngleBetweenTwoPoints2D

// returns the angle between 2 points

// P1: point 1

// P2: point 2

function getAngleBetweenTwoPoints2D(P1, P2) =

let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

    atan2(y2 - y1, x2 - x1);

// getIntersectionPointsOfTwoCircles2D

// returns 2 possible intersection points of 2 circles

// C1: center of circle 1

// r1: radius of circle 1

// C2: center of circle 2

// r2: radius of circle 2

function getIntersectionPointsOfTwoCircles2D(C1, r1, C2, r2) =

let(

    x1=C1[0], y1=C1[1], x2=C2[0], y2=C2[1],

    d=sqrt((x2-x1)^2 + (y2-y1)^2),

    a=(r1^2-r2^2+d^2)/(2*d),

    h=sqrt(r1^2-a^2),

    x3=x1+a*(x2-x1)/d,

    y3=y1+a*(y2-y1)/d,

    x4=x3+h*(y2-y1)/d,

    y4=y3-h*(x2-x1)/d,

    x5=x3-h*(y2-y1)/d,

    y5=y3+h*(x2-x1)/d

) [[x4, y4], [x5, y5]];

---

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I dug into the math a bit more.  I think that yes, the problem is
round-off error.  In the  referenced math you'll note that they don't
compute the final answer as the intersection of two circles, and
neither should you.  Just a slight error in the radius and you go from
a single (tangent) intersection to non-intersecting.  Instead you
should be computing this, as shown in the web site, as the
intersection of the line from M to M1 with circle 1, and then same for
circle 2.

On Fri, Jan 7, 2022 at 8:30 AM Joachim Röttinger joe@joecool.de wrote:

Yes, P1 and P2 have only one intersection point. Function getIntersectionPointsOfTwoCircles2D should return the same point twice.
The circle should intersect and not be inside as the mathematical formulas are correct.
If I take C2 = [50, 0]; it works

-----Ursprüngliche Nachricht-----
Von: Adrian Mariano avm4@cornell.edu
Gesendet: Freitag, 7. Januar 2022 14:21
An: OpenSCAD general discussion discuss@lists.openscad.org
Betreff: [OpenSCAD] Re: Tangential arc between two circles

Your two circles are tangent, with one of them inside the other one.
There is only one intersection point, and the notion of the "tangent arcs" makes no sense in this case.

On Fri, Jan 7, 2022 at 7:31 AM Joachim Röttinger joe@joecool.de wrote:

Hi, I’m trying to write a module that draws an tangential arc between two circles (based on https://www.frassek.org/2d-mathe/kreise-verbinden/).

In principle it works but there are situations where it does not.

The sample code below echos a nan when P2 has been calculated. My assumption is that the issue is based on a loose of precision after calculation of point M.

Can anybody give me a hint how to solve this. Thank you in advance.

$fa = 0.01;

$fs = 0.01;

C1 = [0, 0];

r1 = 20;

C2 = [50, 50];

r2 = 2.5;

R = 90;

test(C1, r1, C2, r2, R);

module test(C1, r1, C2, r2, R) {

R1 = R - r1;

R2 = R - r2;

M = getIntersectionPointsOfTwoCircles2D(C1, R1, C2, R2)[0];

P1 = getIntersectionPointsOfTwoCircles2D(C1, r1, M, R)[0];

P2 = getIntersectionPointsOfTwoCircles2D(C2, r2, M, R)[0];

echo(M, P1, P2);

translate(C1) circle(r=r1);

translate(C2) circle(r=r2);

color("Cyan") polygon(points = [P1, P2, M]);

color("Blue") translate([0, 0, -2]) circleWithCenter2D(r=R, C=M);

polygonWithRadiusBetweenTwoPoints2D(P1, P2, M, r=R);

}

module polygonWithRadiusBetweenTwoPoints2D (P1, P2, C, r=1) {

startAngle = getAngleBetweenTwoPoints2D(C, P1);

endAngle = startAngle + getAngleBetweenThreePoints2D(P1, P2, C);

angles=[ for (i = [startAngle:endAngle-1]) i ];

coords=[ for (j=concat(angles, [endAngle])) [r*cos(j), r*sin(j)]

];

translate(C) polygon(coords);

}

module circleWithCenter2D(r=1, d, C) {

translate(C) circle(r=r);

}

module rotateAboutPoint(a, v, pt) {

translate(pt)

    rotate(a,v)

        translate(-pt)

            children();

}

// getDistanceBetweenTwoPoints2D

// returns the distance between 2 points

// P1: point 1

// P2: point 2

function getDistanceBetweenTwoPoints2D(P1, P2) =

let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

    sqrt((x1 - x2)^2 + (y1 - y2)^2);

// getAngleBetweenThreePoints2D

// returns the angle between 3 points

// P1: point 1

// P2: point 2

// C: center point at angle

function getAngleBetweenThreePoints2D(P1, P2, C) =

let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1], xC=C[0], yC=C[1])

    atan2(y2 - yC, x2 - xC) - atan2(y1 - yC, x1 - xC);

// getAngleBetweenTwoPoints2D

// returns the angle between 2 points

// P1: point 1

// P2: point 2

function getAngleBetweenTwoPoints2D(P1, P2) =

let(x1=P1[0], y1=P1[1], x2=P2[0], y2=P2[1])

    atan2(y2 - y1, x2 - x1);

// getIntersectionPointsOfTwoCircles2D

// returns 2 possible intersection points of 2 circles

// C1: center of circle 1

// r1: radius of circle 1

// C2: center of circle 2

// r2: radius of circle 2

function getIntersectionPointsOfTwoCircles2D(C1, r1, C2, r2) =

let(

    x1=C1[0], y1=C1[1], x2=C2[0], y2=C2[1],

    d=sqrt((x2-x1)^2 + (y2-y1)^2),

    a=(r1^2-r2^2+d^2)/(2*d),

    h=sqrt(r1^2-a^2),

    x3=x1+a*(x2-x1)/d,

    y3=y1+a*(y2-y1)/d,

    x4=x3+h*(y2-y1)/d,

    y4=y3-h*(x2-x1)/d,

    x5=x3-h*(y2-y1)/d,

    y5=y3+h*(x2-x1)/d

) [[x4, y4], [x5, y5]];

---

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---

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try this animation

 try this animation

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