# Plotting an elliptic parabolid

I want to plot the elliptic paraboloid z=4(x-1)^2+(y+2)^2+3 but no matter what the range of x values and the range of y values I give, I can not plot a beautiful ellptic parabolid. This is what I tried

Plot3D[3 + 4 (x-1)^2 + (y+2)^2, {x, -11, 13}, {y, -26, 22}]


and this is what I get

I need to get something similar to this

I even tried to restrict the z values but it did not work.

• Plot3D[3 + 4 (x - 1)^2 + (y + 2)^2, {x, -11, 13}, {y, -26, 22}, RegionFunction -> Function[{x, y, z}, 3 + 4 (x - 1)^2 + (y + 2)^2 < 420]]
– user46676
Mar 17, 2017 at 4:18
• At least closely related: 27169
– Kuba
Mar 17, 2017 at 17:05

Consider using ParametricPlot3D with re-parametrization and translation after, e.g.:

ParametricPlot3D[{u Cos[v]/2, u Sin[v ], u^2} + {1, -2, 3}, {u, 0,
30}, {v, 0, 2 Pi}, BoxRatios -> {1, 1, 1},
PlotRange -> {{-20, 20}, {-35, 35}, {0, 1000}},
ColorFunction -> "Rainbow"]


An alternative approach is

Plot3D[3 + 4 (x - 1)^2 + (y + 2)^2, {x, -20, 20}, {y, -35, 35},
PlotRange -> {0, 1000}, BoxRatios -> {1, 1, .6},
ColorFunction -> "Rainbow", ClippingStyle -> None,
MeshFunctions -> {#3 &, ArcTan[2 #1, #2] &}]


Add the options Axes -> False, Boxed -> False to eliminate the tick marks and box, if desired.

Plot3D[3 + 4 (x - 1)^2 + (y + 2)^2, {x, -20, 20}, {y, -40, 40},
PlotRange -> {0, 1000}, BoxRatios -> {1, 1, .55},
ClippingStyle -> None, Mesh -> {15, 29},
MeshFunctions -> {#3 - 3 &, ArcTan[2 #1 - 2, #2 + 2] &},
ViewPoint -> {0, 100000, 42000}, Axes -> False, Boxed -> False,
ColorFunction -> (Hue[Max[.6 #3/1000 - .4, -.18]] &),
ColorFunctionScaling -> False, Lighting -> {{"Ambient", White}}]


gives an image closer to that in the question. Better agreement probably would require a nonlinear ColorFunction.

• +1 adjusting the plot range and box ratios and nice mesh functions much more natural than my answer...I just like reparametrizations...:) Mar 17, 2017 at 5:53
• Thanks for your answer. I apologize that I can not upvote your answer as I do not have enough reputation to be able to upvote.
– user47406
Mar 17, 2017 at 6:50
• @ubpdqn +1 I like reparameterization too and wish I had thought of it here. Best wishes. Mar 17, 2017 at 19:20