Assignment 7: Saddles and discontinuous surfaces

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Sang Hyun Yun Assignment 7: Saddles and discontinuous surfaces MATH 401 11/8/2021 Partial derivatives and their applications were introduced in multivariable calculus. The usage to find critical points for a two-variable function was the most prominent of these. A crucial point in the domain of a function is a point where the function is either not differentiable or the derivative is zero. A critical point is identified as a local maximum, minimum, or saddle point using the second derivative test. A saddle point is a place on the surface of a function's graph where the slopes/derivatives in orthogonal directions are critical points, but the function is not localized. Saddles come in a variety of shapes and sizes, including plain saddles. Moving from a peak down to a saddle point. The five legs saddle is a variation of the saddle in which the center point is a peak rather than a saddle point. When u is varied from 0 to 2Pi, the result is a saddle-like form. When scaled up to 2kPi, where k is a positive integer bigger than 0, the form generates k copies of itself, intersecting the saddle points. This surface's equation is f(r, t) = a r[cos(n t) - b], where the function has linear r-dependence and is a shifted function of t, and n is the number of waves in the skirt. I put a circular shape on top of the object to make it look like a table shape. Code: a = 0.7; b = 1; dragon = ParametricPlot3D[{r Cos[t], r Sin[t], (a r (Cos[n t] - b)) /. n -> 5}, {r, 0, 40 }, {t, 0, 2 Pi}, PlotPoints -> 50, PlotRange -> All, PlotStyle -> Thickness[1.5], Mesh -> False] f[r_, t_] := {a r (Cos[5 t] - b)}; radius = 1.5; CurvR = Table[{r Cos[u], r Sin[u], f[r, u]}, {u, 0, 2 Pi , 2 Pi/10}]; CurvT = Table[{v Cos[t], v Sin[t], f[v, t]}, {v, 0, 40, 40/10}]; CurvR1 = ParametricPlot3D[{CurvR}, {r, 0, 40}, PlotStyle -> Tube[radius], PlotPoints -> 50, PlotRange -> All]; CurvT1 = ParametricPlot3D[{CurvT}, {t, 0, 2 Pi}, PlotPoints -> 50, PlotStyle -> Tube[radius], PlotRange -> All]; Curv = Show[CurvR1, CurvT1] table = ParametricPlot3D[{r*Sin[Pi/2]*Cos[\[Phi]], r*Sin[Pi/2]*Sin[\[Phi]], r*Cos[Pi/2] 0}, {r, 0, 45}, {\[Phi], 0, 2 Pi}, PlotStyle -> Thickness[2.2], Mesh -> False]; OBS = Show[dragon, table] scale = 25 srad = 15 base1 = RegionPlot3D[(x - 10)^2 + (y - 10)^2 + (z + scale/2 + 13.5)^2 < srad^2 && z > -scale + srad/2, {x, -scale - srad, scale + srad}, {y, -scale - srad, scale + srad}, {z, -scale, scale}, PlotPoints -> 80, Mesh -> False] base2 = RegionPlot3D[(x + 10)^2 + (y - 10)^2 + (z + scale/2 + 13.5)^2 < srad^2 && z > -scale + srad/2, {x, -scale - srad, scale + srad}, {y, -scale - srad, scale + srad}, {z, -scale, scale}, PlotPoints -> 80, Mesh -> False] base3 = RegionPlot3D[(x - 10)^2 + (y + 10)^2 + (z + scale/2 + 13.5)^2 < srad^2 && z > -scale + srad/2, {x, -scale - srad, scale + srad}, {y, -scale - srad, scale + srad}, {z, -scale, scale}, PlotPoints -> 80, Mesh -> False] base4 = RegionPlot3D[(x + 10)^2 + (y + 10)^2 + (z + scale/2 + 13.5)^2 < srad^2 && z > -scale + srad/2, {x, -scale - srad, scale + srad}, {y, -scale - srad, scale + srad}, {z, -scale, scale}, PlotPoints -> 80, Mesh -> False] Final = Show[OBS, Curv, base1, base2, base3, based] Explain: Wolfram Mathematica 12.1 was used to generate this surface. I started by entering the equation for the wavy skirt and setting my parameters to a = 0.7, b = 1, and n = 5 so that the skirt would have five waves. I built the flat surface of the skirt with ParametricPlot3D, then intersected it with a lattice-like model, then I utilized those tables to make two ParametricPlot3D tubular surfaces, which I then pieced together using the Show command to make the whole lattice surface. I used the base function to make support for the object by using RegionPlot3D(Sphere function).