fun with fillets: totally stumped

[bngm]

hi all,

i've got a heck of a problem. the solution will look totally rational, elegant, and logical, but i can't get there to save my life.

i'm trying to fillet two lines, one of which is horizontal and ends at a fixed point. the other starts at a fixed point, but ends somewhere along a fixed axis - not sure where. in other words, the fillet is to take place only within a certain area. the fillet will be tangent to both lines at their endpoints (redundant - that's the definition of "fillet" i suppose). the thing is, i know one of the endpoints in both X and Y for sure. the other endpoint, i only know in its X coordinate. Y is variable. i've shown in blue plines the approximate result in a jpg attached to this post. check the layers to see two of the dozens of attempts i've made to solve the problem graphically.

i'm using rhino but showing you the problem in autocad. the dimensions of all the objects are not real because i'm dealing with tight tolerances in rhino, unrolling surfaces, and i can't afford to approximate. i'm actually pretty surprised that rhino doesn't offer a solution to this problem, none that i've found in the circle or arc dialogues anyway.

i have access to autocad LT 2008, architecture 2008, and rhino 4.

a thousand thanks for any help you can offer!

filletconundrum.dwg

[jaylo23]

Try drawing a circle with a radius of 197.14999952 where the top quadrant of said circle is at the endpoint of the fixed line on the right and let me know if that is what you are looking for.

[jaylo23]

Heres the dwg to be a little more clear.

JAYLO23filletconundrum.dwg

[bngm]

jaylo23, thanks for the reply!

ok, we're almost there. the solution you provided is a very close approximation. i'm looking for the exact perfect result - right now i have two friends working on the solution, and it's becoming apparent that there's always one variable too many to pin down the solution. must use calculus - or have a graphic method included somewhere in our software do it for us!

[Attila The Gel]

Heres how I did it, its more harmonic, but i dont know if its better.

what do you think?

[jaylo23]

I used a combo of trig and calc sprinkled with some geometry. The problem i had is that when working with curves and circles there is always little numbers left over afterwards and it is darn hard to get it down to being perfect, and if this is something that is going to be built and you get it down to a tolerance of say .00000000001 (think i got mine down to .00000001) there is not a machine or person that can build to that tolerance that i know of. But im definitely pulling for you guys and am very intersted in seeing your results and good luck in your endeavors.

[bngm]

attila,

how did you arrive at the red line? we've determined that we need your two magenta lines to be equal. if those two lines are equal, then we can draw the fillet arc and the diagonal line. your magenta lines are identical within 4 significant figures, which is pretty good. the question is how the devil did you find the red one which produced the magenta ones?

i basically would like to find a graphic solution to the problem so that i can continue to find solutions if the dimensions change. this is for a building (a very real one, not a school project) and i predict that we will need to tweak the width of the fillet zone to accommodate a handicapped ramp, among other things.

in addition, it's a pretty rad math problem. funny how they pop up from time to time.

i very humbly thank you all for offering your solutions!

[Attila The Gel]

sorry here it is 2th time I would like to see how you did it

[Attila The Gel]

the green line is the baseline connection between the 2 points that is also the intersection of the lines entering the filletzone. the Red line is simply prependic. from the center of the base wich makes the MAgneta lines equal to perfection. the rest is simple no? circle 3points

[bngm]

the green line is the baseline connection between the 2 points that is also the intersection of the lines entering the filletzone. the Red line is simply prependic. from the center of the base wich makes the MAgneta lines equal to perfection. the rest is simple no? circle 3points

 

i'm still baffled. pleasantly baffled, but baffled nonetheless. here's the problem i encounter: both of the the lines entering the fillet zone are actually not defined, only the horizontal one is. all we know about the diagonal line is its origin on the lower left side. the point where it enters the fillet zone is unknown. that point depends on the radius of the fillet. in fact, both the radius of the fillet and the diagonal entry point depend on each other.

that's how i see it anyway. but you seem to have done it right, and simply.

edit: i'm reading and re-reading your post but cannot replicate your solution. but i also can't find any flaws in your solution - it's totally correct. super nice work, just struggling to figure out how you did it.

[bngm]

so everyone can follow along...

[Attila The Gel]

sorry! this is more detailed!

hope it helps

 

 

Theres a litle flaw!

the angeled magneta line isnt placed right on the end of the other magneta and the red perpendic!

do this and the magnetas are equal

AttilaTheGelfilletconundrum.dwg

[bngm]

thanks eternally for the help. i really appreciate it. you have managed to make the magenta lines equal, and i see how you did it. nice. but it's true that you can make infinitely many isosceles triangles within the fillet zone, it just depends on where you choose to locate the yellow "diagonal entry" point.

the problem is that i still don't believe that first yellow point, the point where the diagonal line enters the fillet zone, can be found by connecting the midpoint of the fillet zone to the origin. this can be proved by checking your solution: draw a line from the origin tangent to the circle. that tangent point is not on the border of the fillet zone.

regardless, this is the same compromise i had reached a couple of days ago, and i will use this midpoint solution to continue the project. the tangent point is really close to the fillet zone border and it lies within the zone anyway, so i'm happy enough and i think i can make it work.

somehow your first solution was correct, i believe the second drawing you posted? the tangent point of the curve was right on the fillet zone border. i still have no idea how you did it.

thank you guys very much for all the help!

for fun and for the math of it, i think it would be extremely cool to have a process that works and produces the absolute correct result every time, like it appears attila's earlier solution did. let's keep bumping the thread.

[Attila The Gel]

the problem is that i still don't believe that first yellow point, the point where the diagonal line enters the fillet zone, can be found by connecting the midpoint of the fillet zone to the origin. this can be proved by checking your solution: draw a line from the origin tangent to the circle. that tangent point is not on the border of the fillet zone.

.

 

Sorry mate but I can't follow this either!

[SEANT]

Here is a solution which employs a common radius.

CommonRadius.dwg

[Attila The Gel]

Here is a solution which employs a common radius.

Truly to simple, nice work!

But I thought fillet meant that when you fillet a corner, the 2 parts are to be equal if you slice from the corner precisely in the middle?

 

nevertheless this is the easiest way! I think

[rkent]

hi all,

i've got a heck of a problem. the solution will look totally rational, elegant, and logical, but i can't get there to save my life.

i'm trying to fillet two lines, one of which is horizontal and ends at a fixed point. the other starts at a fixed point, but ends somewhere along a fixed axis - not sure where. in other words, the fillet is to take place only within a certain area. the fillet will be tangent to both lines at their endpoints (redundant - that's the definition of "fillet" i suppose). the thing is, i know one of the endpoints in both X and Y for sure. the other endpoint, i only know in its X coordinate. Y is variable. i've shown in blue plines the approximate result in a jpg attached to this post. check the layers to see two of the dozens of attempts i've made to solve the problem graphically.

i'm using rhino but showing you the problem in autocad. the dimensions of all the objects are not real because i'm dealing with tight tolerances in rhino, unrolling surfaces, and i can't afford to approximate. i'm actually pretty surprised that rhino doesn't offer a solution to this problem, none that i've found in the circle or arc dialogues anyway.

i have access to autocad LT 2008, architecture 2008, and rhino 4.

a thousand thanks for any help you can offer!

 

Draw a line on top of the one on the left, starting at the lower left and ending at the upper right, end the line command, start the arc command, don't pick a start point, hit enter, now pick the end point to the upper right of your zone.

[rkent]

As far as the original file posted there can't be a solution as the two lines meet at the upper right of the "fillet zone". A fillet or radius tangent to both lines will have to fall outside of the zone, there is no way to have an arc tangent to both lines with these conditions.

The way to determine a center point for the arc tangent to two lines is to draw a line perpendicular from the end points of each line, extending them until they cross. Draw the arc from that intersection. Or measure from the intersection up to the end point of one of the lines and use that value in the fillet command.