Ok, I’ll admit it. I’m stumped. I’m doing some landscaping and I want to smooth out a corner in a walkway. Here’s a site plan (at least as good of one as ASCII art allows):
The lines are the edges of existing sidewalks. I’m rounding out the oblique angle at the bottom and there is a building at the top. I want to lay out an ellipse that is tangential to the bottom and angled sidewalks, preferably at the outside points. I hate doing other people’s homework as much as anyone else on these boards, but is anyone willing to do the math and tell me where to place the focii for an ellipse? I’ve worked on this most of the afternoon and have nothing more than a pile of scrap paper to show for it. I want as gradual of curve as possible, so I assume that one of the focal points will be against the building and the line between the focal points will be parallel to a straight line drawn from the ends of the sidewalk.
Thanks for any help (said while reaching for the aspirin…)
You’re going to run into a problem: any two distinct points you pick on those lines as the points of tangency will determine an ellipse. Which one do you want?
Do you want it to be tangential to the right most sidewalk (the vertical one, as you have drawn it) as well? Do you want the sidewalk to be so that someone walking down (“south”, I’ll say) that right most sidewalk will hit a gradual curve along the ellipse, and follow that curve along to the sidewalk going northwest?
I want the tangents to be at the upper left end of the diagonal sidewalk and the rightmost end of the “bottom” sidewalk. The main traffic flow is from left to right along the “bottom” and diagonal sidewalks.
…But I suspect that an ellipse this big might put a focal point inside the building if the major axis was parallel to the two points, so I might need to change things around a bit.
May I recommend the age old landscaping technique of laying out a piece of rope or garden hose and eyeballin’ it? While it won’t be as mathematically precise, it will probably have a better asthetic.
If you used a circle, you would put the center 51’8" above the lower right corner. I think it would actually give a pretty nice line, with a constant curvature. Any other curve to connect those two points with those tangents would need some section that had higher curvature.
OK, I think I see what you want. The bottom right corner of the sidewalks will still be a right angle, but the bottom left corner will be rounded out with an ellipse, correct?
In that case, I believe this is what you want:
I picked a coordinate system with the origin at the sidewalk corner in the bottom left, and the units are in feet. The ellipse you want is then given by the equation:
Actually, I originally wanted a circle section, but the building at the top of the plan keeps me from doing that ( :smack: sorry, I guess that I could have made a more clearly-worded op.)
Also, this is a volunteer job at my child’s school, and it might be nice in a gee-whiz way to have this as an example of real-life geometry. I haven’t talked with the math teachers yet, but I may throw out the idea of leaving some sort of reference markers at the focal points so that the ellipse can be traced later. Admittedly, this isn’t as much fun as a whisper cove, but it might be a good excuse to have class outside. :shrug:
cornflakes, based on your last response, I assume my solution would be ineffective as well, since the building gets in the way of my proposed foci, also.
I should mention that my solution assumed the ellipse was to be oriented either horizontally or vertically. I haven’t looked at what happens if you allow rotations of such an ellipse; you might be able to get a more workable solution in such a case.
Cabbage, thanks for the help. I’m embarrassed that I didn’t do a better job of pointing out the building in the OP. Yes, I want to smooth out the 153.43 degree* angle between the bottom and diagonal sidewalks, but I need an ellipse with focal points that are more east and west of each other.
Drawing it out on paper, it looks like I might be close (but not quite there) with foci at 32 feet east and west of where the north-south sidewalk meets the building.
*[sub]obtuse, not oblique as I said in the OP.[/sub]
Rotation would work, but it might be nice to have an obvious change in curvature. An offset horizontal orientation would help there (though it will be harder to work around the tree that is in the middle of all of this.)
Oh, my napkin scratches just put the ellipse at (near) the ends of the two sidewalks. I have no idea if the sidewalks are tangential with that.
Folks (esp. Cabbage), thanks for the help, but I blew the setup. The diagonal sidewalk runs at a two to one angle, or 153.43 degrees. The coordinates on the drawing give roughly a 149 degree angle. I was working off drawings I took off the existing sidewalk and confused those with the part to be added. The bottom measurements should be 15 feet on the left and 11-1/2 feet on the right.
My apologies. Out of embarrassment, I’m tempted to ask a Mod to close the thread, but if anyone wants to play with it they’re welcome to.
BTW, putting the foci against the building at +24 feet from the sidewalk on the right makes the ellipse hit both points, but it droops below the line of the LH sidewalk. I may go with this to emphasize the ellipse.
Please don’t close the thread, I’ve just about got some ideas for you. Give me some time to play with the new adjustments you just mentioned–with the new numbers, my original idea may work out so that the foci are not inside the building. Also, I’ve been playing around a bit with rotations, and I’ll post some stuff about that once I make the adjustments.
I don’t know if anything workable will come out of it, but I’ve got a free night and don’t mind playing around with it.
Is there any blocker to the right? I am getting a nice simple ellipse with the center .12 feet right of the right angle and 10.9 feet above it. The foci are 25.79 feet to the right and left of that.
I developed a code that uses nonlinear optimization to find the one ellipse that matches your criterion for any given rotation angle. The best fit seems to use the data I gave above. Tell me if there is any problem. Here is the matlab code if you are interseted.
clear
% a b h k
x0=[10,10,-10,10];
lb=[0,0,-100,-100];
ub=[100,100,100,100];
options='';
for i=-20:10:20
%rotation angle
angle=i*pi/180;
[x,error] = lsqnonlin(@ellipse,x0,lb,ub,options,angle);
y=5;
for j=1:100
x_pos(j)=-26.5+(j-1)/99*26.5;
y=lsqnonlin(@answer,y,-5,20,options,x_pos(j),x,angle);
y_pos(j)=y;
end
plot(x_pos,y_pos)
hold on
end
And the two functions
function F=ellipse(x,angle)
F(1)=x(4)^2/x(2)^2+x(3)^2/x(1)^2-1;
F(2)=(-26.5*sin(angle)+7.5*cos(angle)-x(4))^2/x(2)^2+(-26.5*cos(angle)+7.5*sin(angle)-x(3))^2/x(1)^2-1;
F(3)=tan(angle)+x(2)^2/x(1)^2*x(3)/x(4);
F(4)=(-26.5*sin(angle)+7.5*cos(angle)-x(4))*(sin(angle)-7.5/15*cos(angle))/x(2)^2+(-26.5*cos(angle)+7.5*sin(angle)-x(3))*(cos(angle)-7.5/15*sin(angle))/x(1)^2;
function F=answer(y,x_pos,x,angle)
F=(x_pos*sin(angle)+y*cos(angle)-x(4))^2/x(2)^2+(x_pos*cos(angle)+y*sin(angle)-x(3))^2/x(1)^2-1;
Cabbage, I wouldn’t close the thread. As much as I hate making mistakes, nothing would be accomplished by my burying them. I’m glad that this interests you. Thanks for saying so.
flight, that’s one flat ellipse! Besides the building, there’s a tree about fourteen feet away from the bottom sidewalk, ten feet from the sidewalk on the right. I can work around it (and planned to), but your layout would make things easier.