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+.PS
+# Lyap.m4
+threeD_init
+scale = 1/1.2
+
+viewaz = 30
+viewel = 18
+setview(viewaz,viewel)
+
+Origin: project(0,0,0)
+# Components of view vector W
+w1 = view3D1
+w2 = view3D2
+w3 = view3D3
+# Shape factor of the ellipse on the xy plane
+q = Cos(40)
+
+# cost function
+h = 0.5
+c = 1
+# The projected ellipse is (x/q)^2 + y^2 = c.
+# The cost is v = c+h
+define(`vs',``$2'*q*cos(`$1'),`$2'*sin(`$1')')
+define(`vp',`vs(`$1',`$2'),0')
+define(`vx',`sum3D(vp(`$1',`$2'),0,0,h+(`$2')^2)')
+
+# The gradient of v is (2x/q, 2y, -1) and the line
+# separating front and back is W^T * grad(v) = 0
+# This line intersects the projected ellipse at
+# x1,y1 and x2,y2
+ ap = w2^2*q^2/w1^2+1
+ bp = -w2*w3*q^2/w1^2
+ cp = w3^2*q^2/4/w1^2-c
+ m = sqrt(bp^2-4*ap*cp)
+ y1 = (-bp+m)/ap/2 ; x1 = (w3-2*y1*w2)*q/2/w1
+ y2 = (-bp-m)/ap/2 ; x2 = (w3-2*y2*w2)*q/2/w1
+ t1 = atan2(y1,x1)
+ t2 = atan2(y2,x2)
+ theta1 = min(t1,t2)
+ theta2 = max(t1,t2)
+
+# tangent curve
+ nT = 11
+ for i = 0 to nT do {
+ y = y1 + (y2-y1)/nT*i
+ theta = atan2(y,(w3-2*y*w2)*q/2/w1)
+ r = y/sin(theta)
+ T[i]: project(vx(theta,r))
+ }
+
+# front and back parts of the top curve
+ n = 12
+ for i = 0 to n do {
+ theta = theta1 + (theta2-theta1)/n*i
+ F[i]: project(vx(theta,c))
+ Fp[i]: project(vp(theta,c))
+ }
+ for i = 0 to n do {
+ theta = theta2 + (theta1+twopi_-theta2)/n*i
+ B[i]: project(vx(theta,c))
+ Bp[i]: project(vp(theta,c))
+ }
+
+# trajectory
+rotations = 1.55
+nx = 7
+thetas = 75*dtor_
+thetaf = thetas - rotations*twopi_
+rx = c*0.9
+beta = exp(log(.5)/20)
+
+define(`defX',` rx = `$5' ; np = np-1
+ ts = `$1' ; tf = `$2'
+ for i = 0 to `$3' do {
+ tha = ts + (tf-ts)*i/(`$3')
+ for thx = tha to -twopi_ by twopi_ do {}
+ `$4'[i]: project(vx(thx,rx))
+ Xp[np]: project(vp(thx,rx))
+ np = np+1
+ rx = beta*rx
+ }')
+
+np = 1
+defX(thetas,theta1,nx,X1,rx)
+defX(theta1,theta2-twopi_,nx,X2,rx/beta)
+defX(theta2-twopi_,theta1-twopi_,nx,X3,rx/beta)
+defX(theta1-twopi_,thetaf,5,X4,rx/beta)
+
+# First draw the inside back
+# B is the back curve
+# T is the outline
+ifpstricks(`
+\psset{gradbegin=lightgray,gradend=darkgray,gradlines=1000}
+\pscustom[fillstyle=gradient,gradmidpoint=0.7]{
+ fitcurve(B,n)
+ for i = 0 to nT do {TT[i]: T[nT-i] }
+ fitcurve(TT,nT)
+\relax} ',
+` fitcurve(B,n)
+ for i = 0 to nT do {TT[i]: T[nT-i] }
+ fitcurve(TT,nT) ')
+
+# Centre axis
+thinlines_
+line from Origin to project(0,0,h)
+# F[0] is the leftmost point of the front curve
+line from F[0] to Fp[0]
+# F[n] is the rightmost point of the front curve
+line from F[n] to Fp[n]
+thicklines_
+
+# Now draw the outside front
+ifpstricks(`
+\newgray{gray1}{0.9}%
+\newgray{gray2}{0.4}%
+\psset{gradbegin=gray1,gradend=gray2,gradlines=1000}
+\pscustom[linewidth=0pt,fillstyle=gradient,gradmidpoint=0.99]{
+ fitcurve(F,n)
+ fitcurve(T,nT)
+\relax} ',
+` shade(1,fitcurve(F,n)
+ fitcurve(T,nT)) ')
+# T is the limit curve of visibility
+ fitcurve(T,nT)
+# F is the top front
+ fitcurve(F,n)
+# Front and back projections of the top on xy
+ fitcurve(Fp,n)
+ fitcurve(Bp,n)
+
+# The trajectory in pieces, to allow dashed parts
+ fitcurve(X1,nx)
+ fitcurve(X2,nx,dotted 0.025)
+ fitcurve(X3,nx)
+ fitcurve(X4,3,dotted 0.015)
+ arca(from X4[4] to X4[2],ccw,0.3,<-)
+
+# Projected trajectory
+ np = np-2
+ fitcurve(Xp,np-1)
+ arca(from Xp[np] to Xp[np-2],ccw,0.18,<-)
+ "svg_it(X(t))" at Xp[np]-(2bp__,0) ljust
+
+# Axes and vertical lines
+thinlines_
+ line from X1[0] to Xp[0]
+arrow from Origin to project(1.5,0,0)
+"svg_it(x)`'svg_sub(1)" rjust below
+arrow from Origin to project(0,1.5,0)
+"svg_it(x)`'svg_sub(2)" wid 10bp__ ljust
+line dashed from project(0,0,h) to F[n/2] chop 0 chop arrowht/4
+arrow from F[n/2] to project(0,0,2)
+"svg_it(v(X))" ljust
+
+"svg_it(0)" at Origin+(0,1 pt__) below
+"svg_Omega" at project(0,0.9*c,0) above
+"svg_it(v(X) = c)" at (project(vp(100*dtor_,c)))+(2bp__,0) above ljust
+
+.PE