(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 558195, 19924]*) (*NotebookOutlinePosition[ 559136, 19956]*) (* CellTagsIndexPosition[ 559092, 19952]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData["CONVEXITY OF QUADS"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Luc Barthelet October 1996\ \>", "Section"], Cell[TextData[ "WE HAVE A QUAD WITH CORNERS A,B,C,D. HOW DO WE TEST IF THE QUAD is Convex or \ concave?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We assume that by looking at the intersection of the two lines defined by \ opposite sides of the quad, one can determine if the quad is convex or \ concave. I state that when properly writing the parametric equation of the \ interesection, with the paramters k and t as defined below, the quad is \ convex if the parameters are both of the same sign and outside of the region \ ]0,1[.\nWhen the parameters cannot be calculated, there is an easy way to \ evaluate if the quad is twisted or not."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "Loads the necessary ", StyleBox["Mathematica", FontSlant->"Italic"], " packages" }], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Needs[\"Calculus`VectorAnalysis`\"]"], "Input", AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["First solution"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "Define the 4 points (a,b,c,d) of the quad, and m."], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "a={Xa,Ya};\nb={Xb,Yb};\nc={Xc,Yc};\nd={Xd,Yd};\nm={Xm,Ym};"], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "Define m as a point at in intersection of the lines a-b and c-d"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We write that if m is on the line a-b, there is a value k solution of eq1. \ Same with t solution of eq2, if m is on the line c-d."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["eq1 = m-a==k(b-a);\neq2 = m-c==t(d-c);"], "Input", AspectRatioFixed->True], Cell[TextData[ "This shows we can find k and t by eliminating m from the equations"], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["eq3 = Eliminate[{eq1,eq2}, {Xm,Ym}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ k*Yb == -Ya + k*Ya + Yc - t*Yc + t*Yd && k*Xb == -Xa + k*Xa + Xc - t*Xc + t*Xd && Xb*(-Ya + Yc - t*Yc + t*Yd) == -(Xc*Ya) + t*Xc*Ya - t*Xd*Ya - Xa*Yb + Xc*Yb - t*Xc*Yb + t*Xd*Yb + Xa*Yc - t*Xa*Yc + t*Xa*Yd\ \>", "\<\ k Yb == -Ya + k Ya + Yc - t Yc + t Yd && k Xb == -Xa + k Xa + Xc - t Xc + t Xd && Xb (-Ya + Yc - t Yc + t Yd) == -(Xc Ya) + t Xc Ya - t Xd Ya - Xa Yb + Xc Yb - t Xc Yb + t Xd Yb + Xa Yc - t Xa Yc + t Xa Yd\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["And then solving them for t and k."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Solve[eq3,{t,k}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{k -> -((Ya - Yc)/(-Ya + Yb)) + ((-(Xb*Ya) + Xc*Ya + Xa*Yb - Xc*Yb - Xa*Yc + Xb*Yc)*(Yc - Yd))/ ((-Ya + Yb)*(-(Xc*Ya) + Xd*Ya + Xc*Yb - Xd*Yb + Xa*Yc - Xb*Yc - Xa*Yd + Xb*Yd)), t -> -((-(Xb*Ya) + Xc*Ya + Xa*Yb - Xc*Yb - Xa*Yc + Xb*Yc)/ (-(Xc*Ya) + Xd*Ya + Xc*Yb - Xd*Yb + Xa*Yc - Xb*Yc - Xa*Yd + Xb*Yd))}}\ \>", "\<\ Ya - Yc {{k -> -(--------) + ((-(Xb Ya) + Xc Ya + Xa Yb - Xc Yb - Xa Yc + -Ya + Yb Xb Yc) (Yc - Yd)) / ((-Ya + Yb) (-(Xc Ya) + Xd Ya + Xc Yb - Xd Yb + Xa Yc - Xb Yc - Xa Yd + Xb Yd)), t -> -(Xb Ya) + Xc Ya + Xa Yb - Xc Yb - Xa Yc + Xb Yc -(---------------------------------------------------------------- -(Xc Ya) + Xd Ya + Xc Yb - Xd Yb + Xa Yc - Xb Yc - Xa Yd + Xb Yd )}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "This solutions require divided. We will sho below that we can think of \ solving the equations differently."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Second solution"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Eliminating m from eq1 and eq2 give us:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["eq4 = k(b-a)+a==t(d-c)+c;"], "Input", AspectRatioFixed->True], Cell[TextData["Which we can rewrite as:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["eq5 = k(b-a)+t(c-d)==c-a"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {k*(-Xa + Xb) + t*(Xc - Xd), k*(-Ya + Yb) + t*(Yc - Yd)} == {-Xa + Xc, -Ya + Yc}\ \>", "\<\ {k (-Xa + Xb) + t (Xc - Xd), k (-Ya + Yb) + t (Yc - Yd)} == {-Xa + Xc, -Ya + Yc}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "And if we recognize that we have the product of a 2x2 matrix by a vector, \ the matrix is ", StyleBox["Transpose[{b-a,c-d}]", FontFamily->"Courier", FontWeight->"Bold"], " and the vector is: ", StyleBox["{k,t}", FontFamily->"Courier", FontWeight->"Bold"], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["eq6 = Transpose[{b-a,c-d}].{k,t}==c-a"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {k*(-Xa + Xb) + t*(Xc - Xd), k*(-Ya + Yb) + t*(Yc - Yd)} == {-Xa + Xc, -Ya + Yc}\ \>", "\<\ {k (-Xa + Xb) + t (Xc - Xd), k (-Ya + Yb) + t (Yc - Yd)} == {-Xa + Xc, -Ya + Yc}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["Verify that we have the same thing:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["eq6 === eq5"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["Let's call M the matrix we used."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["M=Transpose[{b-a,c-d}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "We now have a simple equation M.{k,t}== (c-a). If we inverse the matrix M we \ have the solution: {k,t}== Inverse[m].(c-a). This will give us the same \ equations than in the first solution. But since the inverse of the matrix \ requires dividing simple terms by the determinant of the matrix we now can \ see that the determinant factorize the equations above.\n\nThe equation \ becomes: Det[m} {k,t} == Det[M] Inverse[M].(c-a)\n\nWe can check that Det[M] \ Inverse[M] is simple:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Det[M] Inverse[M]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{Yc - Yd, -Xc + Xd}, {Ya - Yb, -Xa + Xb}}\ \>", "\<\ {{Yc - Yd, -Xc + Xd}, {Ya - Yb, -Xa + Xb}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["Det[M] for a 2x2 matrix is simple too:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Collect[Det[M],{Ya,Yb,Yc,Yd}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (Xc - Xd)*Ya + (-Xc + Xd)*Yb + (-Xa + Xb)*Yc + (Xa - Xb)*Yd\ \>", "\<\ (Xc - Xd) Ya + (-Xc + Xd) Yb + (-Xa + Xb) Yc + (Xa - Xb) Yd\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "The introduction suggested that we find k and t and check for their sign and \ verify that their values where outside of ]0,1[. We now change the problem by \ trying to find r and s such as r=Det[M] k and s=Det[M] t, and check for their \ sign and their values to be outside of ]0,Det[M] [."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "{r,s}=Collect[Simplify[Det[M] Inverse[M].(c-a)],{Ya,Yb,Yc,Yd}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {(Xc - Xd)*Ya + (-Xa + Xd)*Yc + (Xa - Xc)*Yd, (-Xb + Xc)*Ya + (Xa - Xc)*Yb + (-Xa + Xb)*Yc}\ \>", "\<\ {(Xc - Xd) Ya + (-Xa + Xd) Yc + (Xa - Xc) Yd, (-Xb + Xc) Ya + (Xa - Xc) Yb + (-Xa + Xb) Yc}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Case where Det[M] = 0"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "If Det[M] = 0 then a-b and c-d are paralelle. We can therefore write \ (b-a)=w(c-d). If w>0 then the quad is convex.\nIf w<0 it is twisted. If w=0 \ it is a triangle."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "If[(Xc-Xd)!=0 , w = (Xb-Xa)/(Xc-Xd),\n\tIf[(Yc-Yd)!=0, w = (Yb-Ya)/(Yc-Yd), \ w=0]\n ];\n "], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Building convexQ"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(* to make the quad follow CW a,b,c,d notation I swapped the input C and D \ in the function definition *)\n\ convexQ[{{Xa_,Ya_},{Xb_,Yb_},{Xd_,Yd_},{Xc_,Yc_}}]:=Module[\n{r,s,det,k,t},\n\ det = (Xc - Xd)*Ya + (-Xc + Xd)*Yb + (-Xa + Xb)*Yc + (Xa - Xb)*Yd;\nIf[det == \ 0,\n\t(* Print[\"det ==0\"]; *)\n\t(* here we cheat, the sign of \ (Xb-Xa)*(Xd-Xc) is the same than the sign of\n\t\t(Xb-Xa)/(Xd-Xc), so we get \ a multiply where was divide *)\n\t\t\n\tIf[(Xd-Xc)!=0 , w = (Xb-Xa)*(Xd-Xc),\n\ \t If[(Yd-Yc)!=0, w = (Yb-Ya)*(Yd-Yc), w=0]\n ];\n TrueQ[w>=0],\n \ \n\t{r,s} = {(Xc - Xd)*Ya + (-Xa + Xd)*Yc + (Xa - Xc)*Yd,\n\t \ (-Xb + Xc)*Ya + (Xa - Xc)*Yb + (-Xa + Xb)*Yc};\n\t \n\t (* \ Print[\"r=\",r];\n\t Print[\"s=\",s];\n\t \ Print[\"det=\",det]; *)\n \n\tIf[det>0, \n\t\tIf[(r \ s<0)||(0r>det)||(0>s>det), False,True]\n\t\t]\n ]\n]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["convexQ[{{0,0},{1,0},{1,1},{0.7,0.6}}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ False\ \>", "\<\ False\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["convexQ[{{0,0},{1,0},{1,1},{0.4,0.6}}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["convexQ[{{0,0},{1,0},{1,1},{0,1}}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ 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