The Utah Teapot in Mathematica

by Jan Mangaldan
hokenjan@yahoo.com

In this notebook, we use Mathematica to render the Utah teapot from a list of vertices and patches. This is then fitted to a Bezier spline surface.

For a brief background on how the Utah teapot came to be, visit The History of The Teapot by Steve Baker.

The Teapot Data

The data for the Utah teapot was obtained from http://sjbaker.org/teapot/teaset.tgz . The original teapot had no bottom, but the dataset from this site has a bottom included.

potPoints = {{1.4, 0, 2.4}, {1.4, -0.784, 2.4}, {0.784, -1.4, 
            2.4}, {0, -1.4, 2.4}, {1.3375, 0, 
            2.53125}, {1.3375, -0.749, 2.53125}, {0.749, -1.3375, 
            2.53125}, {0, -1.3375, 2.53125}, {1.4375, 0, 
            2.53125}, {1.4375, -0.805, 2.53125}, {0.805, -1.4375, 
            2.53125}, {0, -1.4375, 2.53125}, {1.5, 0, 
            2.4}, {1.5, -0.84, 2.4}, {0.84, -1.5, 2.4}, {0, -1.5, 
            2.4}, {-0.784, -1.4, 2.4}, {-1.4, -0.784, 
            2.4}, {-1.4, 0, 2.4}, {-0.749, -1.3375, 
            2.53125}, {-1.3375, -0.749, 2.53125}, {-1.3375, 0, 
            2.53125}, {-0.805, -1.4375, 
            2.53125}, {-1.4375, -0.805, 2.53125}, {-1.4375, 0, 
            2.53125}, {-0.84, -1.5, 2.4}, {-1.5, -0.84, 
            2.4}, {-1.5, 0, 2.4}, {-1.4, 0.784, 2.4}, {-0.784, 
            1.4, 2.4}, {0, 1.4, 2.4}, {-1.3375, 0.749, 
            2.53125}, {-0.749, 1.3375, 2.53125}, {0, 1.3375, 
            2.53125}, {-1.4375, 0.805, 2.53125}, {-0.805, 1.4375, 
            2.53125}, {0, 1.4375, 2.53125}, {-1.5, 0.84, 
            2.4}, {-0.84, 1.5, 2.4}, {0, 1.5, 2.4}, {0.784, 1.4, 
            2.4}, {1.4, 0.784, 2.4}, {0.749, 1.3375, 2.53125}, {1.3375, 
            0.749, 2.53125}, {0.805, 1.4375, 2.53125}, {1.4375, 0.805, 
            2.53125}, {0.84, 1.5, 2.4}, {1.5, 0.84, 2.4}, {1.75, 0, 
            1.875}, {1.75, -0.98, 1.875}, {0.98, -1.75, 
            1.875}, {0, -1.75, 1.875}, {2. , 0, 1.35}, {2. , -1.12, 
            1.35}, {1.12, -2. , 1.35}, {0, -2. , 1.35}, {2. , 0, 
            0.9}, {2. , -1.12, 0.9}, {1.12, -2. , 0.9}, {0, -2. , 
            0.9}, {-0.98, -1.75, 1.875}, {-1.75, -0.98, 
            1.875}, {-1.75, 0, 1.875}, {-1.12, -2. , 
            1.35}, {-2. , -1.12, 1.35}, {-2. , 0, 
            1.35}, {-1.12, -2. , 0.9}, {-2. , -1.12, 
            0.9}, {-2. , 0, 0.9}, {-1.75, 0.98, 1.875}, {-0.98, 
            1.75, 1.875}, {0, 1.75, 1.875}, {-2. , 1.12, 
            1.35}, {-1.12, 2. , 1.35}, {0, 2. , 1.35}, {-2. , 1.12, 
            0.9}, {-1.12, 2. , 0.9}, {0, 2. , 0.9}, {0.98, 1.75, 
            1.875}, {1.75, 0.98, 1.875}, {1.12, 2. , 1.35}, {2. , 1.12, 
            1.35}, {1.12, 2. , 0.9}, {2. , 1.12, 0.9}, {2. , 0, 
            0.45}, {2. , -1.12, 0.45}, {1.12, -2. , 
            0.45}, {0, -2. , 0.45}, {1.5, 0, 0.225}, {1.5, -0.84, 
            0.225}, {0.84, -1.5, 0.225}, {0, -1.5, 0.225}, {1.5, 0, 
            0.15}, {1.5, -0.84, 0.15}, {0.84, -1.5, 
            0.15}, {0, -1.5, 0.15}, {-1.12, -2. , 
            0.45}, {-2. , -1.12, 0.45}, {-2. , 0, 
            0.45}, {-0.84, -1.5, 0.225}, {-1.5, -0.84, 
            0.225}, {-1.5, 0, 0.225}, {-0.84, -1.5, 
            0.15}, {-1.5, -0.84, 0.15}, {-1.5, 0, 
            0.15}, {-2. , 1.12, 0.45}, {-1.12, 2. , 0.45}, {0, 2. , 
            0.45}, {-1.5, 0.84, 0.225}, {-0.84, 1.5, 0.225}, {0, 1.5, 
            0.225}, {-1.5, 0.84, 0.15}, {-0.84, 1.5, 0.15}, {0, 1.5, 
            0.15}, {1.12, 2. , 0.45}, {2. , 1.12, 0.45}, {0.84, 1.5, 
            0.225}, {1.5, 0.84, 0.225}, {0.84, 1.5, 0.15}, {1.5, 0.84, 
            0.15}, {-1.6, 0, 2.025}, {-1.6, -0.3, 
            2.025}, {-1.5, -0.3, 2.25}, {-1.5, 0, 
            2.25}, {-2.3, 0, 2.025}, {-2.3, -0.3, 
            2.025}, {-2.5, -0.3, 2.25}, {-2.5, 0, 
            2.25}, {-2.7, 0, 2.025}, {-2.7, -0.3, 
            2.025}, {-3. , -0.3, 2.25}, {-3. , 0, 
            2.25}, {-2.7, 0, 1.8}, {-2.7, -0.3, 
            1.8}, {-3. , -0.3, 1.8}, {-3. , 0, 1.8}, {-1.5, 
            0.3, 2.25}, {-1.6, 0.3, 2.025}, {-2.5, 0.3, 
            2.25}, {-2.3, 0.3, 2.025}, {-3. , 0.3, 2.25}, {-2.7, 
            0.3, 2.025}, {-3. , 0.3, 1.8}, {-2.7, 0.3, 
            1.8}, {-2.7, 0, 1.575}, {-2.7, -0.3, 
            1.575}, {-3. , -0.3, 1.35}, {-3. , 0, 
            1.35}, {-2.5, 0, 1.125}, {-2.5, -0.3, 
            1.125}, {-2.65, -0.3, 0.9375}, {-2.65, 0, 
            0.9375}, {-2. , -0.3, 0.9}, {-1.9, -0.3, 
            0.6}, {-1.9, 0, 0.6}, {-3. , 0.3, 1.35}, {-2.7, 0.3, 
            1.575}, {-2.65, 0.3, 0.9375}, {-2.5, 0.3, 
            1.125}, {-1.9, 0.3, 0.6}, {-2. , 0.3, 0.9}, {1.7, 0, 
            1.425}, {1.7, -0.66, 1.425}, {1.7, -0.66, 0.6}, {1.7, 0, 
            0.6}, {2.6, 0, 1.425}, {2.6, -0.66, 1.425}, {3.1, -0.66, 
            0.825}, {3.1, 0, 0.825}, {2.3, 0, 2.1}, {2.3, -0.25, 
            2.1}, {2.4, -0.25, 2.025}, {2.4, 0, 2.025}, {2.7, 0, 
            2.4}, {2.7, -0.25, 2.4}, {3.3, -0.25, 2.4}, {3.3, 0, 
            2.4}, {1.7, 0.66, 0.6}, {1.7, 0.66, 1.425}, {3.1, 0.66, 
            0.825}, {2.6, 0.66, 1.425}, {2.4, 0.25, 2.025}, {2.3, 0.25, 
            2.1}, {3.3, 0.25, 2.4}, {2.7, 0.25, 2.4}, {2.8, 0, 
            2.475}, {2.8, -0.25, 2.475}, {3.525, -0.25, 
            2.49375}, {3.525, 0, 2.49375}, {2.9, 0, 2.475}, {2.9, -0.15, 
            2.475}, {3.45, -0.15, 2.5125}, {3.45, 0, 2.5125}, {2.8, 0, 
            2.4}, {2.8, -0.15, 2.4}, {3.2, -0.15, 2.4}, {3.2, 0, 
            2.4}, {3.525, 0.25, 2.49375}, {2.8, 0.25, 2.475}, {3.45, 0.15, 
            2.5125}, {2.9, 0.15, 2.475}, {3.2, 0.15, 2.4}, {2.8, 0.15, 
            2.4}, {0, 0, 3.15}, {0, -0.002, 3.15}, {0.002, 0, 
            3.15}, {0.8, 0, 3.15}, {0.8, -0.45, 3.15}, {0.45, -0.8, 
            3.15}, {0, -0.8, 3.15}, {0, 0, 2.85}, {0.2, 0, 
            2.7}, {0.2, -0.112, 2.7}, {0.112, -0.2, 
            2.7}, {0, -0.2, 2.7}, {-0.002, 0, 
            3.15}, {-0.45, -0.8, 3.15}, {-0.8, -0.45, 
            3.15}, {-0.8, 0, 3.15}, {-0.112, -0.2, 
            2.7}, {-0.2, -0.112, 2.7}, {-0.2, 0, 2.7}, {0, 0.002, 
            3.15}, {-0.8, 0.45, 3.15}, {-0.45, 0.8, 3.15}, {0, 0.8, 
            3.15}, {-0.2, 0.112, 2.7}, {-0.112, 0.2, 2.7}, {0, 0.2, 
            2.7}, {0.45, 0.8, 3.15}, {0.8, 0.45, 3.15}, {0.112, 0.2, 
            2.7}, {0.2, 0.112, 2.7}, {0.4, 0, 2.55}, {0.4, -0.224, 
            2.55}, {0.224, -0.4, 2.55}, {0, -0.4, 2.55}, {1.3, 0, 
            2.55}, {1.3, -0.728, 2.55}, {0.728, -1.3, 
            2.55}, {0, -1.3, 2.55}, {1.3, 0, 2.4}, {1.3, -0.728, 
            2.4}, {0.728, -1.3, 2.4}, {0, -1.3, 
            2.4}, {-0.224, -0.4, 2.55}, {-0.4, -0.224, 
            2.55}, {-0.4, 0, 2.55}, {-0.728, -1.3, 
            2.55}, {-1.3, -0.728, 2.55}, {-1.3, 0, 
            2.55}, {-0.728, -1.3, 2.4}, {-1.3, -0.728, 
            2.4}, {-1.3, 0, 2.4}, {-0.4, 0.224, 2.55}, {-0.224, 
            0.4, 2.55}, {0, 0.4, 2.55}, {-1.3, 0.728, 2.55}, {-0.728, 
            1.3, 2.55}, {0, 1.3, 2.55}, {-1.3, 0.728, 2.4}, {-0.728, 
            1.3, 2.4}, {0, 1.3, 2.4}, {0.224, 0.4, 2.55}, {0.4, 0.224, 
            2.55}, {0.728, 1.3, 2.55}, {1.3, 0.728, 2.55}, {0.728, 1.3, 
            2.4}, {1.3, 0.728, 2.4}, {0, 0, 0}, {1.5, 0, 0.15}, {1.5, 0.84, 
            0.15}, {0.84, 1.5, 0.15}, {0, 1.5, 0.15}, {1.5, 0, 0.075}, {1.5, 
            0.84, 0.075}, {0.84, 1.5, 0.075}, {0, 1.5, 0.075}, {1.425, 0, 
            0}, {1.425, 0.798, 0}, {0.798, 1.425, 0}, {0, 1.425, 
            0}, {-0.84, 1.5, 0.15}, {-1.5, 0.84, 0.15}, {-1.5, 0, 
            0.15}, {-0.84, 1.5, 0.075}, {-1.5, 0.84, 
            0.075}, {-1.5, 0, 0.075}, {-0.798, 1.425, 
            0}, {-1.425, 0.798, 0}, {-1.425, 0, 
            0}, {-1.5, -0.84, 0.15}, {-0.84, -1.5, 
            0.15}, {0, -1.5, 0.15}, {-1.5, -0.84, 
            0.075}, {-0.84, -1.5, 0.075}, {0, -1.5, 
            0.075}, {-1.425, -0.798, 0}, {-0.798, -1.425, 
            0}, {0, -1.425, 0}, {0.84, -1.5, 0.15}, {1.5, -0.84, 
            0.15}, {0.84, -1.5, 0.075}, {1.5, -0.84, 
            0.075}, {0.798, -1.425, 0}, {1.425, -0.798, 
            0}};

potPatches = {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 
            16}, {4, 17, 18, 19, 8, 20, 21, 22, 12, 23, 24, 25, 16, 26, 27, 
            28}, {19, 29, 30, 31, 22, 32, 33, 34, 25, 35, 36, 37, 28, 38, 39, 
            40}, {31, 41, 42, 1, 34, 43, 44, 5, 37, 45, 46, 9, 40, 47, 48, 
            13}, {13, 14, 15, 16, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 
            60}, {16, 26, 27, 28, 52, 61, 62, 63, 56, 64, 65, 66, 60, 67, 68, 
            69}, {28, 38, 39, 40, 63, 70, 71, 72, 66, 73, 74, 75, 69, 76, 77, 
            78}, {40, 47, 48, 13, 72, 79, 80, 49, 75, 81, 82, 53, 78, 83, 84, 
            57}, {57, 58, 59, 60, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 
            96}, {60, 67, 68, 69, 88, 97, 98, 99, 92, 100, 101, 102, 96, 103, 
            104, 105}, {69, 76, 77, 78, 99, 106, 107, 108, 102, 109, 110, 
            111, 105, 112, 113, 114}, {78, 83, 84, 57, 108, 115, 116, 85, 
            111, 117, 118, 89, 114, 119, 120, 93}, {121, 122, 123, 124, 125, 
            126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136}, {124, 
            137, 138, 121, 128, 139, 140, 125, 132, 141, 142, 129, 136, 143, 
            144, 133}, {133, 134, 135, 136, 145, 146, 147, 148, 149, 150, 
            151, 152, 69, 153, 154, 155}, {136, 143, 144, 133, 148, 156, 157, 
            145, 152, 158, 159, 149, 155, 160, 161, 69}, {162, 163, 164, 165, 
            166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 
            177}, {165, 178, 179, 162, 169, 180, 181, 166, 173, 182, 183, 
            170, 177, 184, 185, 174}, {174, 175, 176, 177, 186, 187, 188, 
            189, 190, 191, 192, 193, 194, 195, 196, 197}, {177, 184, 185, 
            174, 189, 198, 199, 186, 193, 200, 201, 190, 197, 202, 203, 
            194}, {204, 204, 204, 204, 207, 208, 209, 210, 211, 211, 211, 
            211, 212, 213, 214, 215}, {204, 204, 204, 204, 210, 217, 218, 
            219, 211, 211, 211, 211, 215, 220, 221, 222}, {204, 204, 204, 
            204, 219, 224, 225, 226, 211, 211, 211, 211, 222, 227, 228, 
            229}, {204, 204, 204, 204, 226, 230, 231, 207, 211, 211, 211, 
            211, 229, 232, 233, 212}, {212, 213, 214, 215, 234, 235, 236, 
            237, 238, 239, 240, 241, 242, 243, 244, 245}, {215, 220, 221, 
            222, 237, 246, 247, 248, 241, 249, 250, 251, 245, 252, 253, 
            254}, {222, 227, 228, 229, 248, 255, 256, 257, 251, 258, 259, 
            260, 254, 261, 262, 263}, {229, 232, 233, 212, 257, 264, 265, 
            234, 260, 266, 267, 238, 263, 268, 269, 242}, {270, 270, 270, 
            270, 279, 280, 281, 282, 275, 276, 277, 278, 271, 272, 273, 
            274}, {270, 270, 270, 270, 282, 289, 290, 291, 278, 286, 287, 
            288, 274, 283, 284, 285}, {270, 270, 270, 270, 291, 298, 299, 
            300, 288, 295, 296, 297, 285, 292, 293, 294}, {270, 270, 270, 
            270, 300, 305, 306, 279, 297, 303, 304, 275, 294, 301, 302, 
            271}};

Viewing The Control Points

Set the Boxed option of Graphics3D to False.

SetOptions[Graphics3D, Boxed -> False];

We visualize the control points using the function toPolygons[], which accepts a list of points and a list of patches as its arguments, and returns the corresponding polygons.

toPolygons[pnts_, pats_] := 
      Map[Polygon[
            Flatten[MapAt[Reverse, Part[Partition[#, 2], {1, 3}], {2}], 
              1]] &, Map[NestList[RotateLeft, #, 2] &, 
          Partition[Part[pnts, pats], 8, 4]], {2}]

roughTeapot = 
        Show[Graphics3D[{toPolygons[potPoints, #] & /@ 
                potPatches}]];

We can also show the "parts" of the teapot.

Show[
        GraphicsArray[{{Graphics3D[{toPolygons[potPoints, #] & /@ 
                    Take[potPatches, 4]}, PlotLabel -> "Rim"], 
              Graphics3D[{toPolygons[potPoints, #] & /@ 
                    Take[potPatches, {5, 12}]}, 
                PlotLabel -> "Body"], 
              Graphics3D[{toPolygons[potPoints, #] & /@ 
                    Take[potPatches, {13, 16}]}, 
                PlotLabel -> "Handle"]}, {Graphics3D[{toPolygons[
                        potPoints, #] & /@ Take[potPatches, {17, 20}]}, 
                PlotLabel -> "Spout"], 
              Graphics3D[{toPolygons[potPoints, #] & /@ 
                    Take[potPatches, {21, 28}]}, 
                PlotLabel -> "Lid"], 
              Graphics3D[{toPolygons[potPoints, #] & /@ 
                    Take[potPatches, {29, 32}]}, 
                PlotLabel -> "Bottom"]}}]];

Rendering The Teapot

To render the teapot, we are going to fit a Bezier surface to the dataset.

We first define the Bernstein polynomials (Subscript[B, i, n])(t):=(({{n}, {i}})) (t^i) (((1 - t))^(n - i)).

Bernstein[i_Integer, n_Integer, t_Symbol]  :=  
      Binomial[n, i] (t^i) (((1 - t))^(n - i))

Bernstein[n_Integer, t_Symbol] := 
      Map[Bernstein[#, n, t] &, Range[0, n]]

It is now straightforward to write a routine to generate the parametric equations for the Bezier surface. (The dot product is used in the following implementation for efficiency.)

BezierSurface[pts_, u_, v_] := 
      Module[{dim = Dimensions[pts]}, 
        Bernstein[dim[[1]] - 1, u] . 
          Map[Bernstein[dim[[2]] - 1, v] . # &, pts]]

Using the BezierSurface[] routine, we define UtahTeapot[] which plots the teapot with u×v points along each patch.

UtahTeapot[u_, v_] := 
        Block[{$DisplayFunction = Identity}, 
          ParametricPlot3D[
            Evaluate[Chop[
                    BezierSurface[#, s, 
                      t]] & /@ (Transpose[
                        Partition[Part[potPoints, #], 4]] & /@ 
                    potPatches)], {s, 0, 1}, {t, 0, 1}, Axes -> None, 
            Boxed -> False, PlotPoints -> {u, v}]];

Show[myTeapot = UtahTeapot[12, 12]];

We can show the teapot along with the control points.

Show[
        roughTeapot /. 
          Polygon[l_] :> {AbsoluteThickness[1/2], GrayLevel[1/2], 
              Line[l]}, 
        roughTeapot /. 
          Polygon[l_] :> 
            Map[{AbsolutePointSize[6], Point[#]} &, l], 
        myTeapot];