Thanks so much for characterizing the problem properly in this manner - that's exactly right. I just did a search for torus-line intersections and found some solutions:
http://tog.acm.org/GraphicsGems/gemsii/intersect/inttor.c On Sep 26, 9:11 pm, "David Joyner" <[EMAIL PROTECTED]> wrote: > Maybe I don't understand the question exactly. > > Consider the circle x^2+y^2=1, z=0, and the set S of all points > at a small distance d, say d near 1/100, from this circle. I think > that is a torus, isn't it? > You want a symbolic equation (ie, a function of d) for the points in > the intersection > of this torus with a line? > > Just guessing, I would imagine this could depend on d in a complicated way, > as when d is small you could have 0, 1, 2, 3, or 4 solutions, but when d is > large, it seems to me you could have 0, 1 or 2 solutions. > Maybe it's not so simple? > > On Fri, Sep 26, 2008 at 11:46 PM, Eugene Jhong <[EMAIL PROTECTED]> wrote: > > > Newbie to sage - trying to find a point on a line in 3D that is a > > specified distance from a circle (the line is not coplanar with the > > circle). Here's what I've inputed into sage: > > > var("line_dir_x line_dir_y line_dir_z line_p_x line_p_y line_p_z > > cent_x cent_y cent_z norm_x norm_y norm_z u radius length") > > line_dir = vector([line_dir_x, line_dir_y, line_dir_z]) # direction > > of line > > line_p = vector([line_p_x, line_p_y, line_p_z]) # point on > > the line > > cent = vector([cent_x, cent_y, cent_z]) # center > > of circle > > norm = vector([norm_x, norm_y, norm_z]) # normal to > > plane of circle > > p = line_p + u * line_dir > > diff0 = p - cent > > dist = diff0.dot_product(norm) > > diff1 = diff0 - dist * norm > > sqr_len = diff1.dot_product(diff1) > > closest_point1 = cent + (radius/sqrt(sqr_len))*diff1 > > diff2 = p - closest_point1 > > eq = length**Integer(2) == diff2.dot_product(diff2) > > solve([eq], u) > > > Here's what eq looks like: > > > rleg_len^2 == (-radius*(-norm_z*(norm_z*(line_dir_z*u + line_p_z - > > cent_z) + norm_y*(line_dir_y*u + line_p_y - cent_y) + > > norm_x*(line_dir_x*u + line_p_x - cent_x)) + line_dir_z*u + line_p_z - > > cent_z)/sqrt((-norm_z*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_z*u + line_p_z - cent_z)^2 + (- > > norm_y*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_y*u + line_p_y - cent_y)^2 + (- > > norm_x*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_x*u + line_p_x - cent_x)^2) + > > line_dir_z*u + line_p_z - cent_z)^2 + (-radius*(- > > norm_y*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_y*u + line_p_y - cent_y)/sqrt((- > > norm_z*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_z*u + line_p_z - cent_z)^2 + (- > > norm_y*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_y*u + line_p_y - cent_y)^2 + (- > > norm_x*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_x*u + line_p_x - cent_x)^2) + > > line_dir_y*u + line_p_y - cent_y)^2 + (-radius*(- > > norm_x*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_x*u + line_p_x - cent_x)/sqrt((- > > norm_z*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_z*u + line_p_z - cent_z)^2 + (- > > norm_y*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_y*u + line_p_y - cent_y)^2 + (- > > norm_x*(norm_z*(line_dir_z*u + line_p_z - cent_z) + > > norm_y*(line_dir_y*u + line_p_y - cent_y) + norm_x*(line_dir_x*u + > > line_p_x - cent_x)) + line_dir_x*u + line_p_x - cent_x)^2) + > > line_dir_x*u + line_p_x - cent_x)^2 > > > I try to do the solve but it doesn't seem to terminate after several > > hours. Just looking for any pointers about finding an analytical > > solution to this seemingly simple problem. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
