geometry help
I'm working on some code to build arches of various types.
I have some code working that builds arches using two circles (trivial).
I am now working on a general "four cirlce" arch (sometimes called "loin of
the donkey", or "ogee" arch). My solution looks like it works when the
radii are equal but the angles of intersection (among other things) are
clearly off. If the radii are not equal this becomes quickly evident (even
if they are equal and you zoom in a lot you can see there is some incorrect
overlap).
Anyone good with geometry who can point out my problem?
I also copied two routines from this list to aid me "pie_slice()" and
"point()"
module archSlice(x,y,r,fnNum,angleStarta=0, angleEnda=0, angleStartb=0,
angleEndb=0) {
translate([x,y,0]) {
pie_slice(r,angleStartb,angleEndb,fnNum);
}
translate([-x,y,0]) {
pie_slice(r,angleStarta,angleEnda,fnNum);
}
}
module archDbl(x,y,r,fnNum) {
intersection() {
translate([x,y,0]) {
circle(r, $fn=fnNum);
}
translate([-x,y,0]) {
circle(r, $fn=fnNum);
}
}
}
module arch(x,y,r,fnNum,angleStarta=0, angleEnda=0, angleStartb=0,
angleEndb) {
if (angleStarta == 0 && angleEnda == 0 && angleStartb == 0 && angleEndb
==0) {
difference() {
archDbl(x,y,r,fnNum);
translate([0,-r/2,0]) {
square([r,r],center=true);
}
}
}
else {
archSlice(x,y,r,fnNum,angleStarta, angleEnda, angleStartb, angleEndb);
}
}
module ogee(x,y,r1,r2,fnNum,width) {
angle = asin((r2+x)/(r1+r2)); //my problem is probably from how I use
this angle
x2 = x - (r1+r2)(sin(angle));
y2 = y + (r1+r2)(cos(angle));
difference() {
arch(x,y,r1,fnNum,0,angle,(180-angle),180); //for instance maybe it
shouldn't be 180-angle
arch(x,y,(r1-width),fnNum,0,angle,(180-angle),180);
}
translate([(x2),(y2),0]) {
difference() {
pie_slice((r2+width),(0-angle),0,fnNum);//or maybe it shouldn't
be 0-angle
pie_slice(r2,(0-angle),0,fnNum);
translate([(-2x2),0,0]) {
pie_slice(r2,180,(180+angle),fnNum);
}
}
}
translate([(-x2),(y2),0]) {
difference() {
pie_slice((r2+width),180,(180+angle),fnNum);//or maybe it
shouldn't be 180+angle
pie_slice(r2,180,(180+angle),fnNum);
translate([(2x2),0,0]) {
pie_slice(r2,(0-angle),0,fnNum);
}
}
}
}
//if you change the fourth parameter to not be equal to the third (or zoom
in) you can see
//there is incorrect overlap etc...
ogee(10,0,20,20,100,2);
module point(x,y) translate([x,y]) circle(0.01);
module pie_slice(r, a0, a1, fnNum) {
//$fa = 5;
R = r * sqrt(2) + 1;
intersection() {
circle(r, $fn=fnNum);
hull() {
point(0,0);
for(i = [0:4])
//a = (((4 - i) * a0 + i * a1) / 4);
point(R * cos((((4 - i) * a0 + i * a1) / 4)), R *
sin((((4 - i) * a0 + i * a1) / 4)));
}
}
}
This code:
r1=20;
r2=40;
rotate( [0,0,-45] ){
translate([-r1,0,0]) pie_slice(r1,0,45, 50);
translate([r2,0,0]) mirror( [0,1,0]) mirror( [1,0,0]) pie_slice(r2,0,45,
50);
}
gives a shape:
http://forum.openscad.org/file/n18394/160916-Ogee_math.png
You can go from there.
$ Runsun Pan, PhD $ libs: doctest , faces ( git ), offline doc ( git ), runscad.py ( 2 , git ), synwrite ( 2 ); $ tips: hash ( 2 ), matrix ( 2 , 3 ), sweep ( 2 ), var ( 2 ), lerp , animation ( gif , prodVid , animlib ), precision ( 2 ), xl-control , type , rounded polygon , chfont , tailRecur ( 2, 3 ), isosphere ( 2 ), area , vol/center , RGB , CurvedImg ; $ Apps: rollApp , blockscad , openjscad , on AWS ( pdf )
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That code is infinitely more elegant than what I wrote.
I might need more help later on since I'm not sure how to make it do some
of the things I am thinking about, but I'll play around with it for now and
ask if I need more help.
On Fri, Sep 16, 2016 at 5:19 PM, runsun runsun@gmail.com wrote:
This code:
r1=20;
r2=40;
rotate( [0,0,-45] ){
translate([-r1,0,0]) pie_slice(r1,0,45, 50);
translate([r2,0,0]) mirror( [0,1,0]) mirror( [1,0,0])
pie_slice(r2,0,45,
50);
}
gives a shape:
http://forum.openscad.org/file/n18394/160916-Ogee_math.png
You can go from there.
$ Runsun Pan, PhD $ libs: doctest , faces ( git ), offline doc ( git ),
runscad.py ( 2 , git ), synwrite ( 2 ); $ tips: hash ( 2 ), matrix (
2 , 3 ), sweep ( 2 ), var ( 2 ), lerp , animation ( gif , prodVid , animlib
), precision ( 2 ), xl-control , type , rounded polygon , chfont ,
tailRecur ( 2, 3 ), isosphere ( 2 ), area , vol/center , RGB , CurvedImg ;
$ Apps: rollApp , blockscad , openjscad , on AWS ( pdf )
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The math of your arcs:
Let O be the arc internal opening, t the arc thickness, r1 and r2 the
external radius of the lower and the internal radius of the upper circles,
then:
the center of the lower circle C1 = [ O / 2 + t - r1, 0]
the center of the upper circle C2 = [ r2, r2sin(a)+r1sin(a) ]
the concordance point CP = [C1[0] + r1cos(a), C2[1] - r2sin(a)]
where a = acos((r2 - O/2 - t - r1)/( r1 + r2 )).
This will give you the outer border of the arc. For the inner border, the
angle a is the same.
Ogee_arc.PNG http://forum.openscad.org/file/n18397/Ogee_arc.PNG
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When you say C1[0] and C2[1] do you just mean C1 and C2 or something else?
On Fri, Sep 16, 2016 at 7:03 PM, Ronaldo rcmpersiano@gmail.com wrote:
The math of your arcs:
Let O be the arc internal opening, t the arc thickness, r1 and r2 the
external radius of the lower and the internal radius of the upper circles,
then:
the center of the lower circle C1 = [ O / 2 + t - r1, 0]
the center of the upper circle C2 = [ r2, r2sin(a)+r1sin(a) ]
the concordance point CP = [C1[0] + r1cos(a), C2[1] - r2sin(a)]
where a = acos((r2 - O/2 - t - r1)/( r1 + r2 )).
This will give you the outer border of the arc. For the inner border, the
angle a is the same.
Ogee_arc.PNG http://forum.openscad.org/file/n18397/Ogee_arc.PNG
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C1[0] is the first component fo C1, C2[1] the second of C2. Then:
CP = [ O / 2 + t - r1 + r1cos(a), r1sin(a)]
2016-09-16 20:14 GMT-03:00 Dan Shriver tabbydan@gmail.com:
When you say C1[0] and C2[1] do you just mean C1 and C2 or something else?
On Fri, Sep 16, 2016 at 7:03 PM, Ronaldo rcmpersiano@gmail.com wrote:
The math of your arcs:
Let O be the arc internal opening, t the arc thickness, r1 and r2 the
external radius of the lower and the internal radius of the upper circles,
then:
the center of the lower circle C1 = [ O / 2 + t - r1, 0]
the center of the upper circle C2 = [ r2, r2sin(a)+r1sin(a) ]
the concordance point CP = [C1[0] + r1cos(a), C2[1] - r2sin(a)]
where a = acos((r2 - O/2 - t - r1)/( r1 + r2 )).
This will give you the outer border of the arc. For the inner border, the
angle a is the same.
Ogee_arc.PNG http://forum.openscad.org/file/n18397/Ogee_arc.PNG
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I'm going to play around with both solutions because runsun's is extremely
elegant but might be a huge challange to my brain to get it to do a
generalized ogee.
runsun: I'm trying to move what you did so the bottom is lying on the x
axis and the top is on the y axis. I can get the bottom on the x axis via
a translate with y value: (r2*(cos(45))) but when I try to do something
similar to get the top in line I can't get it right.
Ronaldo: I'm not sure what "first component for C1" is
On Sat, Sep 17, 2016 at 3:31 PM, Ronaldo Persiano rcmpersiano@gmail.com
wrote:
C1[0] is the first component fo C1, C2[1] the second of C2. Then:
CP = [ O / 2 + t - r1 + r1cos(a), r1sin(a)]
2016-09-16 20:14 GMT-03:00 Dan Shriver tabbydan@gmail.com:
When you say C1[0] and C2[1] do you just mean C1 and C2 or something else?
On Fri, Sep 16, 2016 at 7:03 PM, Ronaldo rcmpersiano@gmail.com wrote:
The math of your arcs:
Let O be the arc internal opening, t the arc thickness, r1 and r2 the
external radius of the lower and the internal radius of the upper
circles,
then:
the center of the lower circle C1 = [ O / 2 + t - r1, 0]
the center of the upper circle C2 = [ r2, r2sin(a)+r1sin(a) ]
the concordance point CP = [C1[0] + r1cos(a), C2[1] - r2sin(a)]
where a = acos((r2 - O/2 - t - r1)/( r1 + r2 )).
This will give you the outer border of the arc. For the inner border, the
angle a is the same.
Ogee_arc.PNG http://forum.openscad.org/file/n18397/Ogee_arc.PNG
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C1 is a 2D point; it has two components or coordinates; C1[0] is the first;
C1[1] is the second.
2016-09-17 22:37 GMT-03:00 Dan Shriver tabbydan@gmail.com:
I'm going to play around with both solutions because runsun's is extremely
elegant but might be a huge challange to my brain to get it to do a
generalized ogee.
runsun: I'm trying to move what you did so the bottom is lying on the x
axis and the top is on the y axis. I can get the bottom on the x axis via
a translate with y value: (r2*(cos(45))) but when I try to do something
similar to get the top in line I can't get it right.
Ronaldo: I'm not sure what "first component for C1" is
On Sat, Sep 17, 2016 at 3:31 PM, Ronaldo Persiano rcmpersiano@gmail.com
wrote:
C1[0] is the first component fo C1, C2[1] the second of C2. Then:
CP = [ O / 2 + t - r1 + r1cos(a), r1sin(a)]
2016-09-16 20:14 GMT-03:00 Dan Shriver tabbydan@gmail.com:
When you say C1[0] and C2[1] do you just mean C1 and C2 or something
else?
On Fri, Sep 16, 2016 at 7:03 PM, Ronaldo rcmpersiano@gmail.com wrote:
The math of your arcs:
Let O be the arc internal opening, t the arc thickness, r1 and r2 the
external radius of the lower and the internal radius of the upper
circles,
then:
the center of the lower circle C1 = [ O / 2 + t - r1, 0]
the center of the upper circle C2 = [ r2, r2sin(a)+r1sin(a) ]
the concordance point CP = [C1[0] + r1cos(a), C2[1] - r2sin(a)]
where a = acos((r2 - O/2 - t - r1)/( r1 + r2 )).
This will give you the outer border of the arc. For the inner border,
the
angle a is the same.
Ogee_arc.PNG http://forum.openscad.org/file/n18397/Ogee_arc.PNG
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DanS wrote
Ronaldo: I'm not sure what "first component for C1" is
See https://en.wikibooks.org/wiki/OpenSCAD_User_Manual/General#Vectors
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DanS wrote
I'm trying to move what you did so the bottom is lying on the x
axis and the top is on the y axis. I can get the bottom on the x axis via
a translate with y value: (r2*(cos(45))) but when I try to do something
similar to get the top in line I can't get it right.
Just translate the whole thing with the correct formula:
r1=20;
r2=40;
a =60;
translate( [0, r2*(sin(a)), 0] ){
rotate( [0,0,-a] ){
translate([-r1,0,0]) pie_slice(r1,0,a, 50);
translate([r2,0,0]) mirror( [0,1,0]) mirror( [1,0,0])
pie_slice(r2,0,a, 50);
}
}
$ Runsun Pan, PhD $ libs: doctest , faces ( git ), offline doc ( git ), runscad.py ( 2 , git ), synwrite ( 2 ); $ tips: hash ( 2 ), matrix ( 2 , 3 ), sweep ( 2 , 3 ), var ( 2 ), lerp , animation ( gif , prodVid , animlib ), precision ( 2 ), xl-control , type , rounded polygon , chfont , tailRecur ( 2, 3 ), isosphere ( 2 ), area , vol/center , RGB , CurvedImg ; $ Apps: rollApp , blockscad , openjscad , on AWS ( pdf )
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runsun I think I finally got it correct when I translated by twice the
width of the curve (at least it worked on the one case I tested). Earlier
I was trying trigonometric translations which all failed
On Sun, Sep 18, 2016 at 12:40 PM, runsun runsun@gmail.com wrote:
DanS wrote
I'm trying to move what you did so the bottom is lying on the x
axis and the top is on the y axis. I can get the bottom on the x axis
via
a translate with y value: (r2*(cos(45))) but when I try to do something
similar to get the top in line I can't get it right.
Just translate the whole thing with the correct formula:
r1=20;
r2=40;
a =60;
translate( [0, r2*(sin(a)), 0] ){
rotate( [0,0,-a] ){
translate([-r1,0,0]) pie_slice(r1,0,a, 50);
translate([r2,0,0]) mirror( [0,1,0]) mirror( [1,0,0])
pie_slice(r2,0,a, 50);
}
}
$ Runsun Pan, PhD $ libs: doctest , faces ( git ), offline doc ( git ),
runscad.py ( 2 , git ), synwrite ( 2 ); $ tips: hash ( 2 ), matrix (
2 , 3 ), sweep ( 2 , 3 ), var ( 2 ), lerp , animation ( gif , prodVid ,
animlib ), precision ( 2 ), xl-control , type , rounded polygon , chfont ,
tailRecur ( 2, 3 ), isosphere ( 2 ), area , vol/center , RGB , CurvedImg ;
$ Apps: rollApp , blockscad , openjscad , on AWS ( pdf )
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runsun: I spoke too soon translating by twice the width only works if
r1=20,r2=10,w=1. I'm trying to put the top point for the r2 pie wedge on
the y axis
On Sun, Sep 18, 2016 at 12:40 PM, runsun runsun@gmail.com wrote:
DanS wrote
I'm trying to move what you did so the bottom is lying on the x
axis and the top is on the y axis. I can get the bottom on the x axis
via
a translate with y value: (r2*(cos(45))) but when I try to do something
similar to get the top in line I can't get it right.
Just translate the whole thing with the correct formula:
r1=20;
r2=40;
a =60;
translate( [0, r2*(sin(a)), 0] ){
rotate( [0,0,-a] ){
translate([-r1,0,0]) pie_slice(r1,0,a, 50);
translate([r2,0,0]) mirror( [0,1,0]) mirror( [1,0,0])
pie_slice(r2,0,a, 50);
}
}
$ Runsun Pan, PhD $ libs: doctest , faces ( git ), offline doc ( git ),
runscad.py ( 2 , git ), synwrite ( 2 ); $ tips: hash ( 2 ), matrix (
2 , 3 ), sweep ( 2 , 3 ), var ( 2 ), lerp , animation ( gif , prodVid ,
animlib ), precision ( 2 ), xl-control , type , rounded polygon , chfont ,
tailRecur ( 2, 3 ), isosphere ( 2 ), area , vol/center , RGB , CurvedImg ;
$ Apps: rollApp , blockscad , openjscad , on AWS ( pdf )
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DanS wrote
runsun: I spoke too soon translating by twice the width only works if
r1=20,r2=10,w=1. I'm trying to put the top point for the r2 pie wedge on
the y axis
Well, applying my math to runsun code I got:
loin(opening=25, thickness=2, r1=25, r2=40,$fn=60);
module loin(opening, thickness, r1, r2) {
// right hand half
intersection() {
mirror([-1,0]) half_loin(opening, thickness, r1, r2);
square(2*(r1+r2));
}
// mirroring it
intersection() {
half_loin(opening, thickness, r1, r2);
mirror([-1,0]) square(2*(r1+r2));
}
}
// half arc before cutting
module half_loin(opening, thickness, r1, r2) {
// my math
c1x = opening/2 + thickness - r1;
c1y = 0;
c2x = r2;
a = acos((c2x-c1x)/(r1+r2));
translate([-r1cos(a)-c1x, r1sin(a)])
// runsun's code
rotate( [0,0,-a] ){
translate([-r2,0,0]) pie_arc(r2,a,thickness);
translate([r1,0,0])
mirror( [0,1,0])
mirror( [1,0,0]) pie_arc(r1,a,-thickness);
}
}
module pie_arc(r,a,t)
difference() {
pie_slice(max(r,r+t),0,a,50);
pie_slice(min(r,r+t),0,a,50);
}
module pie_slice(r, a0, a1)
intersection(){
circle(r);
rotate( a0) translate([-2r,0]) square(4r);
rotate(-a1) translate([-2r,0]) square(4r);
}
Note that $fn is a global variable. See the OpenSCAD manual
https://en.wikibooks.org/wiki/OpenSCAD_User_Manual/Other_Language_Features#.24fa.2C_.24fs_and_.24fn
.
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