# Remove a parametrically defined curve from parametrically defined surface

I want to make a sphere with a pattern removed from it so that I can 3D print it and make a stereograhic projection art display. The equation of the curve is a lengthy Fourier series I won't include, but here's what it looks like:

ParametricPlot[snowflake, {t, -Pi, Pi}, PlotRange -> All]

I then take this equation and project it onto a sphere using the known stereographic projection equations:

X[x_, y_] = (2 x)/(1 + x^2 + y^2)
Y[x_, y_] = (2 y)/(1 + x^2 + y^2)
Z[x_, y_] = (x^2 + y^2 - 1)/(x^2 + y^2 + 1)
s1 = {X[snowflake[[1]], snowflake[[2]]], Y[snowflake[[1]], snowflake[[2]]], Z[snowflake[[1]], snowflake[[2]]]}

The result when plotted and thickened:

Show[ParametricPlot3D[s1, {t, -Pi, Pi}, PlotStyle -> Thickness[.015]]]

However, I don't want to print this this object, I want to print a shere with this curve "cut out" of it. Thoughts?

Edit: Upon request here's the equation for snowflake:

{x[t_], y[t_]} = {8917/35 + (493 Cos[t])/8 - 25/39 Cos[2 t] -
16/21 Cos[3 t] - 5/39 Cos[4 t] - 733/29 Cos[5 t] +
17/45 Cos[6 t] - 46/93 Cos[7 t] + 6/5 Cos[8 t] + 1/11 Cos[9 t] +
1/79 Cos[10 t] - 67/18 Cos[11 t] - 4/13 Cos[12 t] -
78/47 Cos[13 t] - 1/384 Cos[14 t] - 5/28 Cos[15 t] -
1/150 Cos[16 t] - 28/33 Cos[17 t] - 24/73 Cos[18 t] -
302/41 Cos[19 t] - 13/28 Cos[20 t] + 54/37 Cos[21 t] -
3/2 Cos[22 t] + 425/29 Cos[23 t] - 53/41 Cos[24 t] -
757/41 Cos[25 t] - 5/8 Cos[26 t] + 29/15 Cos[27 t] -
11/21 Cos[28 t] + 53/31 Cos[29 t] - 528/529 Cos[30 t] +
79/20 Cos[31 t] + 1/6 Cos[32 t] - 25/33 Cos[33 t] +
39/47 Cos[34 t] - 51/89 Cos[35 t] + 13/12 Cos[36 t] -
27/5 Cos[37 t] + 3/8 Cos[38 t] + 63/62 Cos[39 t] +
12/31 Cos[40 t] - 2/21 Cos[41 t] + 18/31 Cos[42 t] -
26/17 Cos[43 t] + 27/44 Cos[44 t] + 39/32 Cos[45 t] +
32/51 Cos[46 t] + 106/43 Cos[47 t] + 1/14 Cos[48 t] +
1/32 Cos[49 t] - 18/29 Cos[50 t] + 38/63 Cos[51 t] +
1/8 Cos[52 t] + 148/89 Cos[53 t] - 7/30 Cos[54 t] -
25/51 Cos[55 t] - 5/27 Cos[56 t] - 8/25 Cos[57 t] +
3/17 Cos[58 t] - 157/98 Cos[59 t] + 10/33 Cos[60 t] +
53/30 Cos[61 t] + 6/17 Cos[62 t] - 21/32 Cos[63 t] +
14/17 Cos[64 t] - 99/40 Cos[65 t] + 5/17 Cos[66 t] +
61/42 Cos[67 t] + 4/15 Cos[68 t] - 109/136 Cos[69 t] +
37/31 Cos[70 t] - 39/28 Cos[71 t] + 18/29 Cos[72 t] +
146/65 Cos[73 t] - 11/34 Cos[74 t] - 35/36 Cos[75 t] -
47/36 Cos[76 t] + 22/31 Cos[77 t] - 26/33 Cos[78 t] -
16/65 Cos[79 t] + 1/10 Cos[80 t] + (505 Sin[t])/4 -
1/32 Sin[2 t] + 3/29 Sin[3 t] - 9/31 Sin[4 t] + 470/23 Sin[5 t] -
50/151 Sin[6 t] - 722/23 Sin[7 t] - 1/9 Sin[8 t] +
7/10 Sin[9 t] - 14/31 Sin[10 t] + 5/7 Sin[11 t] +
1/53 Sin[12 t] + 159/46 Sin[13 t] - 18/47 Sin[14 t] -
7/18 Sin[15 t] - 9/19 Sin[16 t] + 3/14 Sin[17 t] -
13/22 Sin[18 t] + 139/26 Sin[19 t] - 13/38 Sin[20 t] +
114/229 Sin[21 t] + 42/37 Sin[22 t] + 277/24 Sin[23 t] -
5/12 Sin[24 t] + 119/29 Sin[25 t] - 59/40 Sin[26 t] -
22/19 Sin[27 t] + 5/17 Sin[28 t] + 115/28 Sin[29 t] +
67/48 Sin[30 t] + 13/40 Sin[31 t] + 62/123 Sin[32 t] +
17/37 Sin[33 t] + 19/67 Sin[34 t] - 32/17 Sin[35 t] -
26/27 Sin[36 t] - 85/26 Sin[37 t] - 19/21 Sin[38 t] +
5/14 Sin[39 t] + 5/21 Sin[40 t] - 37/24 Sin[41 t] -
11/35 Sin[42 t] - 109/41 Sin[43 t] - 8/37 Sin[44 t] -
1/10 Sin[45 t] + 19/18 Sin[46 t] - 17/7 Sin[47 t] +
15/32 Sin[48 t] + 27/10 Sin[49 t] + 11/23 Sin[50 t] -
28/39 Sin[51 t] + 7/33 Sin[52 t] - 3/25 Sin[53 t] +
7/24 Sin[54 t] + 31/19 Sin[55 t] - 5/23 Sin[56 t] -
16/25 Sin[57 t] - 1/3 Sin[58 t] - 6/49 Sin[59 t] -
10/49 Sin[60 t] - 32/27 Sin[61 t] + 3/16 Sin[62 t] -
2/15 Sin[63 t] - 17/19 Sin[64 t] - 37/21 Sin[65 t] -
6/55 Sin[66 t] + 5/18 Sin[67 t] - 9/29 Sin[68 t] -
25/39 Sin[69 t] - 33/49 Sin[70 t] - 47/17 Sin[71 t] -
5/51 Sin[72 t] + 60/119 Sin[73 t] + 1/19 Sin[74 t] +
16/25 Sin[75 t] - 5/31 Sin[76 t] + 70/33 Sin[77 t] +
11/18 Sin[78 t] - 1/7 Sin[79 t] + 2/17 Sin[80 t], -(43501/145) + (
7835 Cos[t])/61 + 3/7 Cos[2 t] - 4/13 Cos[3 t] + 51/52 Cos[4 t] -
391/18 Cos[5 t] + 48/49 Cos[6 t] - 702/23 Cos[7 t] -
9/22 Cos[8 t] + 11/34 Cos[9 t] - 2/37 Cos[10 t] -
22/25 Cos[11 t] - 2/21 Cos[12 t] + 112/27 Cos[13 t] -
4/23 Cos[14 t] - 11/32 Cos[15 t] + 25/43 Cos[16 t] +
1/41 Cos[17 t] + 1/22 Cos[18 t] + 173/30 Cos[19 t] -
8/19 Cos[20 t] - 77/32 Cos[21 t] - 29/13 Cos[22 t] -
1077/91 Cos[23 t] - 157/41 Cos[24 t] + 132/25 Cos[25 t] -
49/26 Cos[26 t] - 49/31 Cos[27 t] - 13/64 Cos[28 t] -
105/34 Cos[29 t] + 18/23 Cos[30 t] + 8/35 Cos[31 t] +
55/54 Cos[32 t] + 2/21 Cos[33 t] - 3/41 Cos[34 t] -
37/38 Cos[36 t] - 86/21 Cos[37 t] - 37/51 Cos[38 t] +
8/37 Cos[39 t] - 3/10 Cos[40 t] + 8/17 Cos[41 t] -
2/11 Cos[42 t] - 19/7 Cos[43 t] - 11/25 Cos[44 t] +
23/38 Cos[45 t] - 32/41 Cos[46 t] + 328/109 Cos[47 t] -
9/16 Cos[48 t] + 64/47 Cos[49 t] - 8/43 Cos[50 t] -
7/24 Cos[51 t] - 23/41 Cos[52 t] + 23/31 Cos[53 t] -
8/13 Cos[54 t] + 112/67 Cos[55 t] + 1/22 Cos[56 t] +
7/25 Cos[57 t] + 73/74 Cos[58 t] + 11/40 Cos[59 t] +
17/19 Cos[60 t] - 10/11 Cos[61 t] + 13/45 Cos[62 t] +
51/38 Cos[63 t] + 25/37 Cos[64 t] + 62/35 Cos[65 t] +
19/43 Cos[66 t] - 20/47 Cos[67 t] + 1/9 Cos[68 t] +
19/16 Cos[69 t] + 35/47 Cos[70 t] + 76/29 Cos[71 t] +
16/19 Cos[72 t] - 9/46 Cos[73 t] + 11/28 Cos[74 t] +
3/23 Cos[75 t] - 1/20 Cos[76 t] - 51/37 Cos[77 t] -
8/17 Cos[78 t] + 32/29 Cos[79 t] - 18/35 Cos[80 t] - (
3140 Sin[t])/51 + 7/30 Sin[2 t] - 9/17 Sin[3 t] -
11/17 Sin[4 t] - 1085/41 Sin[5 t] - 667/334 Sin[6 t] +
27/29 Sin[7 t] - 23/18 Sin[8 t] - 5/19 Sin[9 t] +
5/31 Sin[10 t] - 53/22 Sin[11 t] + 3/32 Sin[12 t] +
37/23 Sin[13 t] + 16/35 Sin[14 t] - 33/53 Sin[15 t] +
1/9 Sin[16 t] + 1/93 Sin[17 t] - 7/48 Sin[18 t] +
257/40 Sin[19 t] - 16/37 Sin[20 t] - 1/46 Sin[21 t] -
59/29 Sin[22 t] + 628/35 Sin[23 t] - 41/74 Sin[24 t] +
372/25 Sin[25 t] + 20/33 Sin[26 t] - 17/26 Sin[27 t] +
11/32 Sin[28 t] + 30/19 Sin[29 t] + 1/24 Sin[30 t] -
46/9 Sin[31 t] - 8/65 Sin[32 t] + 49/65 Sin[33 t] -
23/39 Sin[34 t] + 3/37 Sin[35 t] - 4/7 Sin[36 t] +
699/127 Sin[37 t] - 22/41 Sin[38 t] - 64/55 Sin[39 t] +
1/23 Sin[40 t] + 18/25 Sin[41 t] - 10/31 Sin[42 t] +
47/46 Sin[43 t] - 14/31 Sin[44 t] - 18/37 Sin[45 t] +
6/7 Sin[46 t] + 85/29 Sin[47 t] + 29/22 Sin[48 t] -
8/13 Sin[49 t] + 9/40 Sin[50 t] + 8/13 Sin[51 t] +
8/33 Sin[52 t] + 8/5 Sin[53 t] + 59/88 Sin[54 t] +
61/81 Sin[55 t] + 43/57 Sin[56 t] - 33/100 Sin[57 t] +
4/15 Sin[58 t] - 61/18 Sin[59 t] - 3/32 Sin[60 t] -
6/61 Sin[61 t] + 1/24 Sin[62 t] - 28/85 Sin[63 t] +
9/17 Sin[64 t] - 127/42 Sin[65 t] + 8/17 Sin[66 t] +
5/17 Sin[67 t] + 11/16 Sin[68 t] + 1/115 Sin[69 t] +
56/43 Sin[70 t] - 90/53 Sin[71 t] + 17/20 Sin[72 t] -
1/24 Sin[73 t] - 5/22 Sin[74 t] + 19/41 Sin[75 t] -
71/59 Sin[76 t] + 5/11 Sin[77 t] - 48/77 Sin[78 t] -
20/13 Sin[79 t] + 11/15 Sin[80 t]};
snowflake = (1/200) {x[t] - 254, y[t] + 300};
• Please post snowflake. Dec 19, 2022 at 6:19
• I've added the definition of snowflake Dec 19, 2022 at 7:20

We use RegionFunction to cut the curve in the parametric domain.

plot = ParametricPlot[snowflake, {t, -Pi, Pi}, PlotRange -> All];
reg = DiscretizeGraphics[plot];
dist = RegionDistance[reg];

X[x_, y_] = (2 x)/(1 + x^2 + y^2);
Y[x_, y_] = (2 y)/(1 + x^2 + y^2);
Z[x_, y_] = (x^2 + y^2 - 1)/(x^2 + y^2 + 1);

cut = ParametricPlot3D[{X[x, y], Y[x, y], Z[x, y]}, {x, -2,
2}, {y, -2, 2}, Mesh -> None, Boxed -> False, Axes -> False,
PlotPoints -> 80, MaxRecursion -> 4,
RegionFunction -> Function[{x, y, z, u, v}, dist@{u, v} >= .02],
PlotTheme -> "ThickSurface"]