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You are given three 3d cooridates.
This makes a triangle in 3d space. The z values may all be three different, so
it may be rotated in almost*) any direction.
I am just interested of the plane; the surface it makes.
Now, we have a forth point, with only x and y values given. I wish to calculate
the z value where this line meets the plane from the triangle (* -> they will
meet, but not necesserily within the triangle)
If possible, I would like an Excel spreadsheet, but just the formula will do.
TIA,
/Tore
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In lugnet.cad.dev, Tore Eriksson wrote:
> You are given three 3d cooridates.
> This makes a triangle in 3d space. The z values may all be three different, so
> it may be rotated in almost*) any direction.
>
> I am just interested of the plane; the surface it makes.
>
> Now, we have a forth point, with only x and y values given. I wish to calculate
> the z value where this line meets the plane from the triangle (* -> they will
> meet, but not necesserily within the triangle)
>
> If possible, I would like an Excel spreadsheet, but just the formula will do.
>
> TIA,
> /Tore
http://astronomy.swin.edu.au/~pbourke/geometry/planeeq/
tells you how to derive the equation of
a plane passing thru the three points
(x1,y1,z1)
(x2,y2,z2)
(x3,y3,z3)
The standard equation of a plane in 3 space is
A.x + B.y + C.z + D = 0 (where the "." means times)
So given three points in space
(x1,y1,z1),
(x2,y2,z2),
(x3,y3,z3)
you can work out the the equation of the plane through
these points (on which your triangle lies) by working out some
matrix determinants which, if this means nothing to you, boils down to
this -
Calculate A,B,C,D from these equations,
A = y1 .(z2 - z3) + y2 .(z3 - z1) + y3 .(z1 - z2)
B = z1 .(x2 - x3) + z2 .(x3 - x1) + z3 .(x1 - x2)
C = x1 .(y2 - y3) + x2 .(y3 - y1) + x3 .(y1 - y2)
- D = x1 .(y2.z3 - y3.z2) + x2.(y3.z1 - y1.z3) + x3.(y1.z2 - y2.z1)
Having worked these out (i.e. this has given you the A,B,C,D in
A.x + B.y + C.z + D = 0) just plug in your fourth point's x and y co-ords
(say x4 and y4) into this equation i.e. let x=x4 and let y=y4
This gives you the equation for z4 that you are looking for viz.
A.x4 + B.y4 +C.z4 +D =0 (where now the only thing you don't know is z4)
rearranging this is just
z4=(-D -A.x4 - B.x4)/C
Then again, it's late :)
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In lugnet.cad.dev, Ronan Webb wrote:
8<------
> z4=(-D -A.x4 - B.x4)/C
>
>
> Then again, it's late :)
Yes, it's very late. I'd better get myself some sleep first, and have a look at
it tomorrow.
Thanks a lot!
/Tore
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In lugnet.cad.dev, Ronan Webb wrote:
> In lugnet.cad.dev, Tore Eriksson wrote:
> > You are given three 3d cooridates.
>
> (snip) <
> > If possible, I would like an Excel spreadsheet, but just the formula will do.
> >
> > TIA,
> > /Tore
>
>
>
> http://astronomy.swin.edu.au/~pbourke/geometry/planeeq/
>
> (snip) <
> Having worked these out (i.e. this has given you the A,B,C,D in
> A.x + B.y + C.z + D = 0) just plug in your fourth point's x and y co-ords
> (say x4 and y4) into this equation i.e. let x=x4 and let y=y4
> This gives you the equation for z4 that you are looking for viz.
> A.x4 + B.y4 +C.z4 +D =0 (where now the only thing you don't know is z4)
> rearranging this is just
> z4=(-D -A.x4 - B.x4)/C
>
>
> Then again, it's late :)
Hey! cool! where was this info when I wrote my calculator program? It would have made it alot easier. The calculator grew into a slightly different purpose though. it was for helping determine where to trim lines and quads at intersection points.
The last formula had a typo, here it is corrected.
z4=(-D -A.x4 - B.y4)/C
Tore, I will email you the spreadsheet I put this information into.
Paul
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In lugnet.cad.dev, Paul Easter wrote:
> In lugnet.cad.dev, Ronan Webb wrote:
> > In lugnet.cad.dev, Tore Eriksson wrote:
> > > You are given three 3d cooridates.
> >
> > (snip) <
> > > If possible, I would like an Excel spreadsheet, but just the formula will do.
> > >
> > > TIA,
> > > /Tore
> >
> >
> >
> > http://astronomy.swin.edu.au/~pbourke/geometry/planeeq/
> >
> > (snip) <
> > Having worked these out (i.e. this has given you the A,B,C,D in
> > A.x + B.y + C.z + D = 0) just plug in your fourth point's x and y co-ords
> > (say x4 and y4) into this equation i.e. let x=x4 and let y=y4
> > This gives you the equation for z4 that you are looking for viz.
> > A.x4 + B.y4 +C.z4 +D =0 (where now the only thing you don't know is z4)
> > rearranging this is just
> > z4=(-D -A.x4 - B.x4)/C
> >
> >
> > Then again, it's late :)
>
> Hey! cool! where was this info when I wrote my calculator program? It would have made it alot easier. The calculator grew into a slightly different purpose though. it was for helping determine where to trim lines and quads at intersection points.
>
> The last formula had a typo, here it is corrected.
> z4=(-D -A.x4 - B.y4)/C
Yes, I saw that, too.
> Tore, I will email you the spreadsheet I put this information into.
>
> Paul
Thanks for the offer, but I have already made a working spreadsheet. :)
Now, all I have to do is try to automate it with some VBS.
/Tore
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In lugnet.cad.dev, Paul Easter wrote:
> In lugnet.cad.dev, Ronan Webb wrote:
> > In lugnet.cad.dev, Tore Eriksson wrote:
> > > You are given three 3d cooridates.
> >
> > (snip) <
> > > If possible, I would like an Excel spreadsheet, but just the formula will do.
> > >
> > > TIA,
> > > /Tore
> >
> >
> >
> > http://astronomy.swin.edu.au/~pbourke/geometry/planeeq/
> >
> > (snip) <
> > Having worked these out (i.e. this has given you the A,B,C,D in
> > A.x + B.y + C.z + D = 0) just plug in your fourth point's x and y co-ords
> > (say x4 and y4) into this equation i.e. let x=x4 and let y=y4
> > This gives you the equation for z4 that you are looking for viz.
> > A.x4 + B.y4 +C.z4 +D =0 (where now the only thing you don't know is z4)
> > rearranging this is just
> > z4=(-D -A.x4 - B.x4)/C
> >
> >
> > Then again, it's late :)
>
> Hey! cool! where was this info when I wrote my calculator program? It would have made it alot easier. The calculator grew into a slightly different purpose though. it was for helping determine where to trim lines and quads at intersection points.
>
> The last formula had a typo, here it is corrected.
> z4=(-D -A.x4 - B.y4)/C
>
> Tore, I will email you the spreadsheet I put this information into.
>
> Paul
Sorry guys, just wanted to store this function in a safer place than my hard
drive. :)
I just don't want to remake it again...!
Feel free to copy/paste/edit/port if you need it!
/Tore
// calc_z4 by Tore Eriksson
// credits to Ronan Webb for the superb formula
double calc_z4(double x1,double y1,double z1,
double x2,double y2,double z2,
double x3,double y3,double z3,
double x4,double y4)
{
double a,b,c,minus_d,z4;
a = y1*(z2-z3) + y2*(z3-z1) + y3*(z1-z2);
b = z1*(x2-x3) + z2*(x3-x1) + z3*(x1-x2);
c = x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2);
minus_d = x1*(y2*z3-y3*z2) + x2*(y3*z1-y1*z3) + x3*(y1*z2-y2*z1);
z4 = (minus_d - a*x4 - b*y4)/c;
return z4;
}
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> // calc_z4 by Tore Eriksson
> // credits to Ronan Webb for the superb formula
> double calc_z4(double x1,double y1,double z1,
> double x2,double y2,double z2,
> double x3,double y3,double z3,
> double x4,double y4)
> {
> double a,b,c,minus_d,z4;
> a = y1*(z2-z3) + y2*(z3-z1) + y3*(z1-z2);
> b = z1*(x2-x3) + z2*(x3-x1) + z3*(x1-x2);
> c = x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2);
> minus_d = x1*(y2*z3-y3*z2) + x2*(y3*z1-y1*z3) + x3*(y1*z2-y2*z1);
> z4 = (minus_d - a*x4 - b*y4)/c;
> return z4;
> }
You also need a touch of code to catch when c=0 so that z4 is undefined ;)
Tim
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In lugnet.cad.dev, Timothy Gould wrote:
// calc_z4 by Tore Eriksson
// credits to Ronan Webb for the superb formula
double calc_z4(double x1,double y1,double z1,
double x2,double y2,double z2,
double x3,double y3,double z3,
double x4,double y4)
{
double a,b,c,minus_d,z4;
a = y1*(z2-z3) + y2*(z3-z1) + y3*(z1-z2);
b = z1*(x2-x3) + z2*(x3-x1) + z3*(x1-x2);
c = x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2);
if(c==0)
{
err=11;
c=1;
}
minus_d = x1*(y2*z3-y3*z2) + x2*(y3*z1-y1*z3) + x3*(y1*z2-y2*z1);
z4 = (minus_d - a*x4 - b*y4)/c;
return z4;
}
>
> You also need a touch of code to catch when c=0 so that z4 is undefined ;)
>
> Tim
That shouldn't really be happening, unless someone enters a "triangle" that
isn't a triangle, for example with x1, x2, and x3 all = 0, or a line that isn't
referring to a polygon for some reason yet is processed as one.
Hmm, division by zero is not a thing to play with. I might add these 3 LOC's
anyway. Thanks for pointing this possible/hypothetical source of errors out.
/Tore
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In lugnet.cad.dev, Tore Eriksson wrote:
> That shouldn't really be happening, unless someone enters a "triangle" that
> isn't a triangle, for example with x1, x2, and x3 all = 0, or a line that isn't
> referring to a polygon for some reason yet is processed as one.
Actually, if the original triangle is (for example) in the Y-Z plane, then just
passing in an x4 that's different from x1, x2, and x3 will result in a failure.
The equivalent goes for an initial triangle in the X-Z plane.
--Travis
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In lugnet.cad.dev, Travis Cobbs wrote:
> In lugnet.cad.dev, Tore Eriksson wrote:
> > That shouldn't really be happening, unless someone enters a "triangle" that
> > isn't a triangle, for example with x1, x2, and x3 all = 0, or a line that isn't
> > referring to a polygon for some reason yet is processed as one.
>
> Actually, if the original triangle is (for example) in the Y-Z plane, then just
> passing in an x4 that's different from x1, x2, and x3 will result in a failure.
> The equivalent goes for an initial triangle in the X-Z plane.
>
> --Travis
True, but such triangles will not be visible in the 2-D front view of the UI and
therefor not clickable - unless the program is buggy. And if you can't click on
them, the function never has to encounter data from them. Hopefully...
/Tore
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