I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.
(A full circle is 2*pi*r) I also need to find the point on an arc that is X
distance away.

So far, I have a messy solution involving sines and cosines with IF
statements for which quadranttoo determine the fraction of a full circle.
There must be an elegant solution.

Background on what I'm doing: I'm writing a custom Gcode for my CNC control
to read the Z distance on a laser displacement sensor every given increment
along the arc.

"Karl Townsend" wrote in message
anews.com...
I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.
(A full circle is 2*pi*r) I also need to find the point on an arc that is X
distance away.

So far, I have a messy solution involving sines and cosines with IF
statements for which quadranttoo determine the fraction of a full
circle. There must be an elegant solution.

Background on what I'm doing: I'm writing a custom Gcode for my CNC
control to read the Z distance on a laser displacement sensor every given
increment along the arc.


When you say "start point and end point" do you mean the length of the
chord? Do you know the angle?

When you say "start point and end point" do you mean the length of the
chord? Do you know the angle?

Start point is an X, Y coordinate
End point is a second X, Y coordinate
Arc center is a third X, Y coordinate.

You can find the angle to a point on the arc by Arcsine(deltaX/radius).
radius is the distance between either start point or end point and arc
center.

Karl

s=r*angle in radians
YOu have the x,y of each point.
Assume the lowest pt in y is at 0 degrees (to the right of the center)
The angle can be deduced from :
Dy (height, or y distance) = r * sin(angle), so
angle = 1/arcsin (Dy) -You'll need trig tables or a sci. calculator.

Distance along arc, then, is r*angle (degrees/57.3)

This was my basic approach. But the sine function gives the angle within a
given quadrant. Then, i had to test each angle for which quadrant to know
how many 90 degrees to add. The result is a total mess. I'm hoping for a
more elegant approach.

Karl

Karl Townsend wrote:
I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.
(A full circle is 2*pi*r) I also need to find the point on an arc that is X
distance away.

So far, I have a messy solution involving sines and cosines with IF
statements for which quadranttoo determine the fraction of a full circle.
There must be an elegant solution.

Background on what I'm doing: I'm writing a custom Gcode for my CNC control
to read the Z distance on a laser displacement sensor every given increment
along the arc.


s=r*angle in radians
YOu have the x,y of each point.
Assume the lowest pt in y is at 0 degrees (to the right of the center)
The angle can be deduced from :
Dy (height, or y distance) = r * sin(angle), so
angle = 1/arcsin (Dy) -You'll need trig tables or a sci. calculator.

Distance along arc, then, is r*angle (degrees/57.3)
Now, distance along the arc is not the same as distance away.
Dx^2 + Dy^2 = h^2, where h is the distance away.
So, square up the increment in x, same in y, add 'em together
then take the sqrt of that number.
And it's not 7am here yet, so I could be all screwed up / mark

On Thu, 12 Mar 2009 08:16:11 -0500, Karl Townsend wrote:

I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.
(A full circle is 2*pi*r) I also need to find the point on an arc that is X
distance away.

So far, I have a messy solution involving sines and cosines with IF
statements for which quadranttoo determine the fraction of a full circle.
There must be an elegant solution.

Background on what I'm doing: I'm writing a custom Gcode for my CNC control
to read the Z distance on a laser displacement sensor every given increment
along the arc.

I cannot see how this is hard, as written.

You know the radius - it's distance between one end and the centre. You
want to know what angle from that line will give you X along the arc,
right? That's X over 2*pi*radius in radians - do the math and convert to
degrees.

The above from memory of grade-school trig some decades ago; contents may
have settled/become dis-ordered.

On Mar 12, 9:16*am, "Karl Townsend"
wrote:
I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.

"Karl Townsend" wrote in message
anews.com...
I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.
(A full circle is 2*pi*r) I also need to find the point on an arc that is X
distance away.

So far, I have a messy solution involving sines and cosines with IF
statements for which quadranttoo determine the fraction of a full
circle. There must be an elegant solution.

Background on what I'm doing: I'm writing a custom Gcode for my CNC
control to read the Z distance on a laser displacement sensor every given
increment along the arc.

Don't know if this is what you're looking for, but it worked for me. My
situation is that I had to calculate the actual distance of curbing along
both sides of a 2,000 long street. The distance is going to be different
along the outside of the street and the inside of the street. Both
distances were going to be different than a straight line from start to end
of street. It didn't mean a lot of difference mathematically, but my boss
was impressed that I figured out how to calculate it so accurately. (Google
was not mentioned, just that I worked hard at achieving the results.)

BTW, sagitta is a word that comes in there somewhere.

Play with it and see. It allows you to enter variables that spits out the
unknown.

http://www.1728.com/circsect.htm

HTH

Steve

Karl Townsend wrote:
I need help with math and geometry theory. I need to know how to calculate
the distance along an arc given the start point, end point, and arc center.
(A full circle is 2*pi*r) I also need to find the point on an arc that is X
distance away.

So far, I have a messy solution involving sines and cosines with IF
statements for which quadranttoo determine the fraction of a full circle.
There must be an elegant solution.

Background on what I'm doing: I'm writing a custom Gcode for my CNC control
to read the Z distance on a laser displacement sensor every given increment
along the arc.








Arc length when you know the chord length and the radius:


Chord Length = SQRT( (X0-X1)^2 + (Y0-Y1)^2 )

Angle in Radians = 2 * ARCSIN( (Chord Length) / (2 * Arc Radius) )

Arc Length = Arc Radius * Angle in Radians

--
Steve Walker
(remove wallet to reply)

Karl Townsend writes:

I need help with math and geometry theory. I need to know how to
calculate the distance along an arc given the start point, end point,
and arc center.

This is harder than anyone thinks because the system may be rotated.

It will work to "change coordinates" (*) to put the arc center at (0,0),
and the start point at some (x,0). Then the coordinates of the end point
will be at a known angle by an inverse trig function of the second point's
coordinates. Divide the angle by 2pi radians to get the fraction of the
circumference of the given radius.

You can also calculate the angle directly with vector arithmetic.

The book _Graphics Gems_ is a cookbook for this stuff:

http://www.acm.org/tog/GraphicsGems/

(*) Change coordinates = that stuff you never learned in algebra class
because short skirts were in style.

"Richard J Kinch" wrote in message
. ..
Karl Townsend writes:

I need help with math and geometry theory. I need to know how to
calculate the distance along an arc given the start point, end point,
and arc center.

This is harder than anyone thinks because the system may be rotated.

I'm glad somebody thought it wasn't that easy. I got a good solution by
brute force, but don't know the theory behind it. I made 12 equal spaced
radii at 30 degree intervals around the circle. For each point, I recorded
the sine ratio, sin result from the computer, cosine ratio, cosine result
from the computer. (Tan gives divide by 0 errors)

By inspection I can see:
IF delta Y (point-arc center) =0 then cos(deltaX/radius)
gives the result

IF delta Y 0 then (360 - cos(deltaX/radius) gives the result.

Do this for both points, then subtract to get the angle. Works for any
angle, any point in space. Piece of cake. of course getting it programmed
into the CNC control will take effort.

Karl

On Fri, 13 Mar 2009 17:40:17 -0500, "Karl Townsend"
wrote:


"Richard J Kinch" wrote in message
...
Karl Townsend writes:

I need help with math and geometry theory. I need to know how to
calculate the distance along an arc given the start point, end point,
and arc center.

This is harder than anyone thinks because the system may be rotated.

I'm glad somebody thought it wasn't that easy. I got a good solution by
brute force, but don't know the theory behind it. I made 12 equal spaced
radii at 30 degree intervals around the circle. For each point, I recorded
the sine ratio, sin result from the computer, cosine ratio, cosine result
from the computer. (Tan gives divide by 0 errors)

By inspection I can see:
IF delta Y (point-arc center) =0 then cos(deltaX/radius)
gives the result

IF delta Y 0 then (360 - cos(deltaX/radius) gives the result.

Do this for both points, then subtract to get the angle. Works for any
angle, any point in space. Piece of cake. of course getting it programmed
into the CNC control will take effort.

Karl


I don't know what programming language you're using but look for a two
argument arctangent function in the list of available math functions.

In C, it's atan2(y,x).

These functions will return the proper angle/quadrant, regardless of
the signs of the arguments, x and y. Some examples...

atan2(1,0) = 90 degrees
atan2(0,-1) = 180
atan2(-1,0) = 270
atan2(-1,-1) = 225

I don't know what programming language you're using but look for a two
argument arctangent function in the list of available math functions.

In C, it's atan2(y,x).
....

Shoot, I knew there was an easy way. My control language doesn't have
this function. It has the ability to call a small C function that you
write. Now that I'm nearly done, it would be more work to do this.

Thanks for the tip.

Karl

On Fri, 13 Mar 2009 18:48:35 -0500, Karl Townsend
wrote:


I don't know what programming language you're using but look for a two
argument arctangent function in the list of available math functions.

In C, it's atan2(y,x).
...

Shoot, I knew there was an easy way. My control language doesn't have
this function. It has the ability to call a small C function that you
write. Now that I'm nearly done, it would be more work to do this.

Thanks for the tip.

Karl

There is a wiki page for that too:

http://en.wikipedia.org/wiki/Atan2

May give you some info to muse over...

--
Leon Fisk
Grand Rapids MI/Zone 5b
Remove no.spam for email

"Karl Townsend" wrote in
anews.com:

I need help with math and geometry theory. I need to know
how to calculate the distance along an arc given the start
point, end point, and arc center. (A full circle is 2*pi*r)
I also need to find the point on an arc that is X distance
away.

So far, I have a messy solution involving sines and cosines
with IF statements for which quadranttoo determine the
fraction of a full circle. There must be an elegant
solution.

Background on what I'm doing: I'm writing a custom Gcode
for my CNC control to read the Z distance on a laser
displacement sensor every given increment along the arc.



http://www.handymath.com/cgi-bin/arc18.cgi?submit=Entry

Ken