# # Test cases for SSI. # # Gershon Elber, Feb 1995 # # In order to be able to appreciate the complexity of some of the test cases, # it is suggested to view this file through IRIT with a display device that # is able to render the surfaces shaded, such as xgldrvs. # # # 1. Simple close loop intersection. Both srfs are bi-quadratic polynomials. # s1 = sbezier( list( list( ctlpt( E3, 0.1, 0.0, 1.0 ), ctlpt( E3, 0.3, 1.0, 0.5 ), ctlpt( E3, 0.0, 2.0, 1.0 ) ), list( ctlpt( E3, 1.1, 0.0, 0.5 ), ctlpt( E3, 1.3, 1.0, 0.0 ), ctlpt( E3, 1.0, 2.0, 0.5 ) ), list( ctlpt( E3, 2.1, 0.0, 1.1 ), ctlpt( E3, 2.3, 1.0, 0.4 ), ctlpt( E3, 2.0, 2.0, 1.2 ) ) ) ); s2 = s1 * scale( vector( 1, 1, -1 ) ) * tz( 1.2 ); color( s1, red ); color( s2, green ); All = list( s1, s2 ); interact( All ); save( "ssi1", All ); # # 2 and 3. Same as 1 but both surfaces are degree raised. How much does it # slows down ssi computation!? # s1a = sraise( sraise( s1, ROW, 4 ), COL, 4 ); s2a = sraise( sraise( s2, ROW, 4 ), COL, 4 ); color( s1a, red ); color( s2a, green ); All = list( s1a, s2a ); interact( All ); save( "ssi2", All ); s1b = sraise( sraise( s1a, ROW, 5 ), COL, 5 ); s2b = sraise( sraise( s2a, ROW, 5 ), COL, 5 ); color( s1b, red ); color( s2b, green ); All = list( s1b, s2b ); interact( All ); save( "ssi3", All ); # # 4. Two biquadratic polynomial Bspline surfaces. Intersection is open. # s1 = sbspline( 3, 3, list( list( ctlpt( E3, 0.1, 0.0, 1.0 ), ctlpt( E3, 0.3, 1.0, 0.5 ), ctlpt( E3, 0.0, 2.0, 1.0 ) ), list( ctlpt( E3, 1.1, 0.0, 0.5 ), ctlpt( E3, 1.3, 1.0, 0.0 ), ctlpt( E3, 1.0, 2.0, 0.5 ) ), list( ctlpt( E3, 2.1, 0.0, 1.1 ), ctlpt( E3, 2.3, 1.0, 0.4 ), ctlpt( E3, 2.0, 2.0, 1.2 ) ), list( ctlpt( E3, 3.1, 0.0, 1.9 ), ctlpt( E3, 3.3, 1.1, 1.4 ), ctlpt( E3, 3.0, 2.0, 1.9 ) ), list( ctlpt( E3, 4.1, 0.0, 1.1 ), ctlpt( E3, 4.3, 1.0,-0.4 ), ctlpt( E3, 4.0, 2.2, 1.2 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ); s2 = sbspline( 3, 3, list( list( ctlpt( E3, 0.1, 0.0, 1.0 ), ctlpt( E3, 0.3, 0.7, 0.5 ), ctlpt( E3, 0.1, 1.2, 1.0 ), ctlpt( E3, 0.0, 2.0, 0.5 ) ), list( ctlpt( E3, 1.1, 0.0, 0.5 ), ctlpt( E3, 1.3, 1.0, 0.0 ), ctlpt( E3, 1.1, 1.3, 0.5 ), ctlpt( E3, 1.0, 2.0, 0.5 ) ), list( ctlpt( E3, 2.1, 0.0, 1.1 ), ctlpt( E3, 2.3, 0.5, 0.4 ), ctlpt( E3, 2.0, 1.0, 1.3 ), ctlpt( E3, 2.0, 2.0, 0.4 ) ), list( ctlpt( E3, 3.1, 0.0, 1.9 ), ctlpt( E3, 3.3, 0.7, 1.4 ), ctlpt( E3, 3.1, 1.1, 1.5 ), ctlpt( E3, 3.1, 2.0, 1.9 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ) * rotx( 90 ) * trans( vector( 1.5, 1.5, 0 ) ); color( s1, red ); color( s2, green ); All = list( s1, s2 ); interact( All ); save( "ssi4", All ); # # Two biquadratic rational surface intersection - a cone and a sphere. # s1 = coneSrf( 3.0, 0.7 ); s2 = sphereSrf( 0.8 ) * trans( vector( 0.35, 0.65, 1.3 ) ); color( s1, red ); color( s2, green ); All = list( s1, s2 ); interact( All ); save( "ssi5", All ); # # Same as 5, but the poles of the sphere are on the cone's surface. # s1 = coneSrf( 3.0, 0.7 ); s2 = sphereSrf( 0.8 ) * roty( -atan2( 0.7, 3.0 ) * 180 / Pi ) * trans( vector( 0.35, 0.0, 1.5 ) ); color( s1, red ); color( s2, green ); All = list( s1, s2 ); interact( All ); save( "ssi6", All ); # # Four different and isolated intersection loops between two bicubic # Bspline surfaces. # s1 = sbspline( 4, 4, list( list( ctlpt( E3, 0.1, 0.0, 0.51 ), ctlpt( E3, 0.4, 1.0, 0.52 ), ctlpt( E3, 0.2, 2.2, 0.5 ), ctlpt( E3, 0.4, 3.5, 0.49 ), ctlpt( E3, 0.0, 4.3, 0.52 ) ), list( ctlpt( E3, 1.1, 0.3, 0.53 ), ctlpt( E3, 1.3, 1.3, 2.7 ), ctlpt( E3, 1.1, 2.2,-1.4 ), ctlpt( E3, 1.1, 3.3, 3.1 ), ctlpt( E3, 1.2, 4.2, 0.48 ) ), list( ctlpt( E3, 2.1, 0.1, 0.47 ), ctlpt( E3, 2.4, 1.1, 0.52 ), ctlpt( E3, 2.3, 2.0, 0.51 ), ctlpt( E3, 2.4, 3.3, 0.52 ), ctlpt( E3, 2.0, 4.0, 0.53 ) ), list( ctlpt( E3, 3.1, 0.4, 0.49 ), ctlpt( E3, 3.3, 1.3, 2.6 ), ctlpt( E3, 2.9, 2.1,-1.9 ), ctlpt( E3, 2.9, 3.5, 2.0 ), ctlpt( E3, 3.0, 4.6, 0.51 ) ), list( ctlpt( E3, 4.1, 0.1, 0.53 ), ctlpt( E3, 4.0, 1.2, 0.45 ), ctlpt( E3, 4.3, 2.0, 0.51 ), ctlpt( E3, 3.9, 3.4, 0.55 ), ctlpt( E3, 4.0, 4.2, 0.51 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ); s2 = sbspline( 4, 4, list( list( ctlpt( E3, 0.1, 0.0, 1.85 ), ctlpt( E3, 0.4, 1.0, 1.9 ), ctlpt( E3, 0.2, 2.2, 1.95 ), ctlpt( E3, 0.4, 3.5, 1.7 ), ctlpt( E3, 0.0, 4.3, 1.8 ) ), list( ctlpt( E3, 1.1, 0.3, 1.88 ), ctlpt( E3, 1.3, 1.3,-1.1 ), ctlpt( E3, 1.1, 2.2, 2.85 ), ctlpt( E3, 1.1, 3.3,-0.95 ), ctlpt( E3, 1.2, 4.2, 1.7 ) ), list( ctlpt( E3, 2.1, 0.1, 1.9 ), ctlpt( E3, 2.4, 1.1, 1.8 ), ctlpt( E3, 2.3, 2.0, 1.85 ), ctlpt( E3, 2.4, 3.3, 1.65 ), ctlpt( E3, 2.0, 4.0, 1.75 ) ), list( ctlpt( E3, 3.1, 0.4, 1.85 ), ctlpt( E3, 3.3, 1.3,-0.9 ), ctlpt( E3, 2.9, 2.1, 2.4 ), ctlpt( E3, 2.9, 3.5,-0.9 ), Ctlpt( E3, 3.0, 4.6, 1.8 ) ), list( ctlpt( E3, 4.1, 0.1, 1.85 ), ctlpt( E3, 4.0, 1.2, 1.75 ), ctlpt( E3, 4.3, 2.0, 1.65 ), ctlpt( E3, 3.9, 3.4, 1.95 ), ctlpt( E3, 4.0, 4.2, 1.85 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ); color( s1, red ); color( s2, green ); All = list( s1, s2 ); interact( All ); save( "ssi7", All ); # # Same as 7 but we scale the Z axis by 0.1 and by 0.01 to create almost # tangent surfaces. # s1a = s1 * sz( 0.1 ); s2a = s2 * sz( 0.1 ); color( s1a, red ); color( s2a, green ); All = list( s1a, s2a ); interact( All ); save( "ssi8", All ); s1b = s1 * sz( 0.01 ); s2b = s2 * sz( 0.01 ); color( s1b, red ); color( s2b, green ); All = list( s1b, s2b ); interact( All ); save( "ssi9", All ); # # Two different intersection curves that are very close to each other. # Intersection between two biquadratic Bezier saddle like surfaces. # In the last example of this sequence, the surfaces are tangent at # the center point, s1( 0.5, 0.5 ) = s2( 0.5, 0.5 ) ~= (1.175, 1.13, 1.49 ) # s1 = sbezier( list( list( ctlpt( E3, 0.1, 0.0, 1.6 ), ctlpt( E3, 0.3, 1.1, 0.4 ), ctlpt( E3, 0.0, 2.2, 1.5 ) ), list( ctlpt( E3, 1.1, 0.2, 3.0 ), ctlpt( E3, 1.3, 1.0, 1.4 ), ctlpt( E3, 1.0, 2.2, 2.7 ) ), list( ctlpt( E3, 2.1, 0.1, 1.4 ), ctlpt( E3, 2.3, 1.3, 0.2 ), ctlpt( E3, 2.0, 2.2, 1.2 ) ) ) ); p = seval( s1, 0.5, 0.5 ); s2a = s1 * trans( -coerce( p, vector_type ) ) * scale( vector( 1.2, 1.1, -0.5 ) ) * rotz( 15 ) * trans( vector( coord( p, 1 ), coord( p, 2 ), coord( p, 3 ) + 0.1 ) ); color( s1, red ); color( s2a, green ); All = list( s1, s2a ); interact( All ); save( "ssi10", All ); s2b = s1 * trans( -coerce( p, vector_type ) ) * scale( vector( 1.2, 1.1, -0.5 ) ) * rotz( 15 ) * trans( vector( coord( p, 1 ), coord( p, 2 ), coord( p, 3 ) + 0.01 ) ); color( s1, red ); color( s2b, green ); All = list( s1, s2b ); interact( All ); save( "ssi11", All ); s2c = s1 * trans( -coerce( p, vector_type ) ) * scale( vector( 1.2, 1.1, -0.5 ) ) * rotz( 15 ) * trans( vector( coord( p, 1 ), coord( p, 2 ), coord( p, 3 ) + 0.001 ) ); color( s1, red ); color( s2c, green ); All = list( s1, s2c ); interact( All ); save( "ssi12", All ); s2d = s1 * trans( -coerce( p, vector_type ) ) * scale( vector( 1.2, 1.1, -0.5 ) ) * rotz( 15 ) * trans( vector( coord( p, 1 ), coord( p, 2 ), coord( p, 3 ) ) ); color( s1, red ); color( s2d, green ); All = list( s1, s2d ); interact( All ); save( "ssi13", All ); free( p ); # # Another case of almost tangency. Here we have a fairly flat surface and # an elliptic surface almots tangent (first case), tangent at a point (second # case) and intersects in a tiny loop (third case). Both happens at the # centers of the surfaces # s1( 1.5, 1.0 ) ~= s2( 1.0, 1.5 ) ~= ( 2.28, 2.14, 0.48 ) # Both surfaces are cubic by quadratic Bspline surfaces. # s1 = sbspline( 3, 4, list( list( ctlpt( E3, 0.1, 0.0, 0.3 ), ctlpt( E3, 0.4, 1.0, 0.1 ), ctlpt( E3, 0.2, 2.2, 0.5 ), ctlpt( E3, 0.4, 3.5, 0.1 ), ctlpt( E3, 0.0, 4.3, 0.2 ) ), list( ctlpt( E3, 1.1, 0.3, 0.15 ), ctlpt( E3, 1.3, 1.3, 0.3 ), ctlpt( E3, 1.1, 2.2, 0.2 ), ctlpt( E3, 1.1, 3.3, 0.25 ), ctlpt( E3, 1.2, 4.2, 0.0 ) ), list( ctlpt( E3, 2.1, 0.1, 0.4 ), ctlpt( E3, 2.4, 1.1, 0.7 ), ctlpt( E3, 2.3, 2.0, 0.35 ), ctlpt( E3, 2.4, 3.3, 0.22 ), ctlpt( E3, 2.0, 4.0, 0.35 ) ), list( ctlpt( E3, 3.1, 0.4, 0.11 ), ctlpt( E3, 3.3, 1.3, 0.1 ), ctlpt( E3, 2.9, 2.1, 0.2 ), ctlpt( E3, 2.9, 3.5, 0.3 ), Ctlpt( E3, 3.0, 4.6, 0.3 ) ), list( ctlpt( E3, 4.1, 0.1, 0.12 ), ctlpt( E3, 4.0, 1.2, 0.05 ), ctlpt( E3, 4.3, 2.0, 0.33 ), ctlpt( E3, 3.9, 3.4, 0.13 ), ctlpt( E3, 4.0, 4.2, 0.27 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ); s2 = sbspline( 4, 3, list( list( ctlpt( E3, 0.1, 0.0, 1.85 ), ctlpt( E3, 0.4, 1.0, 1.9 ), ctlpt( E3, 0.2, 2.2, 1.95 ), ctlpt( E3, 0.4, 3.5, 1.7 ), ctlpt( E3, 0.0, 4.3, 1.8 ) ), list( ctlpt( E3, 1.1, 0.3, 1.88 ), ctlpt( E3, 1.3, 1.3, 1.6 ), ctlpt( E3, 1.1, 2.2, 0.7 ), ctlpt( E3, 1.1, 3.3, 1.5 ), ctlpt( E3, 1.2, 4.2, 1.7 ) ), list( ctlpt( E3, 2.1, 0.1, 1.9 ), ctlpt( E3, 2.4, 1.1, 0.5 ), ctlpt( E3, 2.3, 2.0, 0.1 ), ctlpt( E3, 2.4, 3.3, 0.5 ), ctlpt( E3, 2.0, 4.0, 1.75 ) ), list( ctlpt( E3, 3.1, 0.4, 1.85 ), ctlpt( E3, 3.3, 1.3, 1.3 ), ctlpt( E3, 2.9, 2.1, 0.5 ), ctlpt( E3, 2.9, 3.5, 1.4 ), Ctlpt( E3, 3.0, 4.6, 1.8 ) ), list( ctlpt( E3, 4.1, 0.1, 1.85 ), ctlpt( E3, 4.0, 1.2, 1.75 ), ctlpt( E3, 4.3, 2.0, 1.65 ), ctlpt( E3, 3.9, 3.4, 1.95 ), ctlpt( E3, 4.0, 4.2, 1.85 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ); # # Compute a rotation matrix that rotates s2 so that its tangent plane at # (0.5, 0.5) is the same as the tangent plane of s1 at (0.5, 0.5). # nrml1 = normalize( coerce( seval( snrmlsrf( s1 ), 1.5, 1.0 ), vector_type ) ); tan1a = normalize( coerce( seval( sderive( s1, ROW ), 1.5, 1.0 ), vector_type ) ); tan1b = tan1a ^ nrml1; rot1 = homomat( list( list( coord( tan1a, 0 ), coord( tan1a, 1 ), coord( tan1a, 2 ), 0 ), list( coord( tan1b, 0 ), coord( tan1b, 1 ), coord( tan1b, 2 ), 0 ), list( coord( nrml1, 0 ), coord( nrml1, 1 ), coord( nrml1, 2 ), 0 ), list( 0, 0, 0, 1 ) ) ); free( tan1a ); free( tan1b ); free( nrml1 ); nrml2 = normalize( coerce( seval( snrmlsrf( s2 ), 1.0, 1.5 ), vector_type ) ); tan2a = normalize( coerce( seval( sderive( s2, ROW ), 1.0, 1.5 ), vector_type ) ); tan2b = tan2a ^ nrml2; rot2 = homomat( list( list( coord( tan2a, 0 ), coord( tan2a, 1 ), coord( tan2a, 2 ), 0 ), list( coord( tan2b, 0 ), coord( tan2b, 1 ), coord( tan2b, 2 ), 0 ), list( coord( nrml2, 0 ), coord( nrml2, 1 ), coord( nrml2, 2 ), 0 ), list( 0, 0, 0, 1 ) ) ); free( tan2a ); free( tan2b ); free( nrml2 ); RotMat = rot2^-1 * rot1; free( rot1 ); free( rot2 ); # # Apply the rotation matrix. # s2r = s2 * rotmat; free( RotMat ); # # Prove it: here are the normals of both surfaces at (0.5, 0.5). # normalize( coerce( seval( snrmlsrf( s1 ), 1.5, 1.0 ), vector_type ) ); normalize( coerce( seval( snrmlsrf( s2r ), 1.0, 1.5 ), vector_type ) ); pt1 = seval( s1, 1.5, 1.0 ); pt2 = seval( s2r, 1.0, 1.5 ); s2a = s2r * trans( vector( coord( pt1, 1 ) - coord( pt2, 1 ), coord( pt1, 2 ) - coord( pt2, 2 ), coord( pt1, 3 ) - coord( pt2, 3 ) + 0.01 ) ); color( s1, red ); color( s2a, green ); All = list( s1, s2a ); interact( All ); save( "ssi14", All ); s2b = s2r * trans( vector( coord( pt1, 1 ) - coord( pt2, 1 ), coord( pt1, 2 ) - coord( pt2, 2 ), coord( pt1, 3 ) - coord( pt2, 3 ) ) ); color( s1, red ); color( s2b, green ); All = list( s1, s2b ); interact( All ); save( "ssi15", All ); s2c = s2r * trans( vector( coord( pt1, 1 ) - coord( pt2, 1 ), coord( pt1, 2 ) - coord( pt2, 2 ), coord( pt1, 3 ) - coord( pt2, 3 ) - 0.01 ) ); color( s1, red ); color( s2c, green ); All = list( s1, s2c ); interact( All ); save( "ssi16", All ); free( s2r ); free( pt1 ); free( pt2 ); # # A complex but single intersection curve. This is between two quadratic # Bspline surfaces. # s1 = sbspline( 3, 3, list( list( ctlpt( E3, 0.1, 0.0, 0.51 ), ctlpt( E3, 0.4, 1.0, 0.52 ), ctlpt( E3, 0.2, 2.2, 0.5 ), ctlpt( E3, 0.4, 3.5, 0.49 ), ctlpt( E3, 0.0, 4.3, 0.52 ) ), list( ctlpt( E3, 1.1, 0.3, 0.53 ), ctlpt( E3, 1.3, 1.3, 1.7 ), ctlpt( E3, 1.1, 2.2,-0.4 ), ctlpt( E3, 1.1, 3.3, 1.1 ), ctlpt( E3, 1.2, 4.2, 0.48 ) ), list( ctlpt( E3, 2.1, 0.1, 0.47 ), ctlpt( E3, 2.4, 1.1, 1.52 ), ctlpt( E3, 2.3, 2.0, 0.51 ), ctlpt( E3, 2.4, 3.3, 1.52 ), ctlpt( E3, 2.0, 4.0, 0.53 ) ), list( ctlpt( E3, 3.1, 0.4, 0.49 ), ctlpt( E3, 3.3, 1.3, 1.6 ), ctlpt( E3, 2.9, 2.1,-0.9 ), ctlpt( E3, 2.9, 3.5, 1.0 ), ctlpt( E3, 3.0, 4.6, 0.51 ) ), list( ctlpt( E3, 4.1, 0.1, 0.53 ), ctlpt( E3, 4.0, 1.2, 0.45 ), ctlpt( E3, 4.3, 2.0, 0.51 ), ctlpt( E3, 3.9, 3.4, 0.55 ), ctlpt( E3, 4.0, 4.2, 0.51 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ); s2 = sbspline( 3, 3, list( list( ctlpt( E3, 0.1, 0.0, 1.45 ), ctlpt( E3, 0.4, 1.0, 1.5 ), ctlpt( E3, 0.2, 2.2, 1.55 ), ctlpt( E3, 0.4, 3.5, 1.3 ), ctlpt( E3, 0.0, 4.3, 1.4 ) ), list( ctlpt( E3, 1.1, 0.3, 1.48 ), ctlpt( E3, 1.3, 1.3,-0.5 ), ctlpt( E3, 1.1, 2.2, 1.45 ), ctlpt( E3, 1.1, 3.3,-0.5 ), ctlpt( E3, 1.2, 4.2, 1.3 ) ), list( ctlpt( E3, 2.1, 0.1, 2.5 ), ctlpt( E3, 2.4, 1.1, 2.4 ), ctlpt( E3, 2.3, 2.0, 0.1 ), ctlpt( E3, 2.4, 3.3, 2.4 ), ctlpt( E3, 2.0, 4.0, 2.25 ) ), list( ctlpt( E3, 3.1, 0.4, 1.45 ), ctlpt( E3, 3.3, 1.3,-0.5 ), ctlpt( E3, 2.9, 2.1, 1.0 ), ctlpt( E3, 2.9, 3.5,-0.3 ), Ctlpt( E3, 3.0, 4.6, 1.4 ) ), list( ctlpt( E3, 4.1, 0.1, 1.45 ), ctlpt( E3, 4.0, 1.2, 1.35 ), ctlpt( E3, 4.3, 2.0, 1.25 ), ctlpt( E3, 3.9, 3.4, 1.55 ), ctlpt( E3, 4.0, 4.2, 1.45 ) ) ), list( list( KV_OPEN ), list( KV_OPEN ) ) ) * tz( -0.26 ); color( s1, red ); color( s2, green ); All = list( s1, s2 ); interact( All ); save( "ssi17", All ); free( All ); free( s1 ); free( s2 ); free( s1a ); free( s1b ); free( s2a ); free( s2b ); free( s2c ); free( s2d );