Parametric Surface Puns

https://i.imgur.com/P3hcOHX.jpg

When i parametricise it (not sure what the correct term is) i get the correct answer of 9pi. but when i dont parametricise it, (which i prefer doing, i dont like parametrics) which seems like it should work, my surface integral just becomes 2(x^2 + y^2)dxdy, the zdS turns into 2dxdy. which cant give me 9pi. am i missing something.

I'm trying to create a table with the attached design -- but instead of just flat 1x2 slats I want to do a parametric plywood design similar to attached video. Any tips on achieving this on the cone shaped base?

Basically a semi-organic shape would wrap around the cone shaped base of this table -- attached to an upper and lower metal ring.

https://preview.redd.it/27k304fyn7881.png?width=2192&format=png&auto=webp&s=3dd3a8cadb2e9c8ac7cc2c1404e1be5fd7839f32

https://preview.redd.it/8yrwmybpn7881.png?width=1936&format=png&auto=webp&s=982418780d63b6d72df5da43adb08d936f953700

So suppose we have a surface in 3-space, and are given the generalized parametric from of the surface as R(u,v)=(x(u,v), y(u,v), z(u,v)) and also has a level surface f(x,y,z)=1. Show that grad(f) is perpendicular to the tangent vectors (partial derivatives) dR/du and dR/dv.

I was given a hint that the question is an application of the chain rule and implicit differentiation.

I started by first taking the general derivatives to obtain (all partials) dR/du=(dx(u,v)/du, dy(u,v)/du, dz(u,v)/du), likewise for dR/dv, and grad(f)=(df(x,y,z)/dx, df(x,y,z)/dy, df(x,y,z)/dz). And the idea that perpendicular vectors have a dot product of 0.

Naturally I began with taking the dot product of the two

=dx(u,v)/du * df(x,y,z)/dx +dy(u,v)/du *df(x,y,z)/dy+dz(u,v)/du*df(x,y,z)/dz

As is this does not simplify to 0. So I begin exploring the idea of implicit differentiation. The problem i arrive at is if I implicitly derive, for example, df(x,y,z)/dx, I end up with more partials regarding x,y, or z. and doing so for dx(u,v)/du we get more partials regarding u and v. Nothing seems to cancel since the parametric form is not in terms of x,y,z variables and f(x,y,z) is not in terms of u,v variables. I just get longer, more complex equations that do not cancel.

I then took to the idea that the cross product of dR/du and dR/dv should be parallel (or opposite parallel) to the gradient, but again I arrive at a long string of partials that do not cancel and are of 5 different variables.

I have been on this for almost 9 hours now and have made no progress.

Small tutorial of how to skirt a network of sketches with a Nurbs surface which allows for modifications without rebuilding

https://youtu.be/o_6jUmLkIM8

https://math.stackexchange.com/questions/461547/whats-the-equation-of-helix-surface

I don't understand the last steps in the last answer in this link... How do we go from the binormal, normal and tangent vectors to the surface area equation?

Thanks!

https://www.youtube.com/watch?v=8jEqGdDhq6I

Free response question for x^(2/3)+y^(2/3)=1 or x=cos^3 (t) and y=sin^3 (t)

) Sketch the curve

) Find the length

) Find the area

) Find the surface area

) Find the volume

Parametric questions seem almost always related to PVA (position, velocity, acceleration) and so everything but the first two sub-questions wouldn't be on the AP tests. The ideas behind the last 3 questions is to show how parametric is just another way of looking at rectangular equations. Obviously this is after the AP tests of 2021, but calculus knowledge and practice can still be useful if taking it in college.

How would one implement a differential form in manim? If one was to plot a parametric surface, a certain number of tangent vectors would have to be drawn on the plot corresponding to the rows in the Jacobian for the coordinate patch. Orthogonality would be assumed as the vectors would need to span the tangent space of the k-dimensional manifold. Is there existing functionality in either the community repository or the manimgl branch? I am concerned less with drawing the tangent lines and more with updating them for various points on the manifold. Of course, I could simply fabricate continuous data for the visualizations, yet I do not think it would be as satisfying. Is there a method to visualize a fibre bundle on a manifold using manim?

This is the definition of a parametric surface we were given: https://i.imgur.com/ETPyxOm.jpg

Now the question is: let f : R - > R^4 be a map given by f(t) = ( cos(t), sin(t), cos(at), sin(at) ), where a is irrational. Let Gamma be the curve traced out by the image of f, as t is varied over all reals. Prove that Gamma is not a parametric surface.

How would one even go about doing this? I'm really not sure where to start. If I had to prove it was a param. surf., I'd just look for a parametrization and show that it satisfies the properties it needs to. Trying to show that such a thing can't be done at all seems much more difficult.

Any help or tips would be greatly appreciated!

#fintech #algotrading #hedgefunds #quants #hft A new arbitrage-free parametric volatility surface CLICK HERE TO VIEW THE PDF Alexandre Antonov, Michael Spector and Michael Konikov describe a new parametric volatility surface that is arbitrage free, is extremely rich and flexible, and has closed-form expressions for both European option values and local volatilities. The volatility surface is based on previous work by Carr and Pelts, for which the present authors provide a simple derivation and a concrete implementation Parametric volatility surfaces are used in finance and financial

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https://reddit.com/link/i6fseh/video/ar5oxglfoxf51/player

Parametric curve x = acos(t), y= bsin(t) Find length of the curve from t = 0 to t = pi/2 (I dunno what to do with the variables a and b)

Find area of the surface about the Y-AXIS

y = ∫ UB = x, LB = 1 | sqrt(sqrt(t) - 1) dt | [1<=x<=16]

(Do I just plug in x for t and find the derivative like normal?)

3. find ds/dx

   s(x) = ∫ UB = x, LB = 2 | sqrt(1 + e^t) dt

   (Is number 3 just plugging x for t?)

i want to add vector over the surface of sphere after creating sphere through parametric function

How do I find the parametric UV's of a flat surface; compared to that of sphere ? Is the parametric UV's of the flat surface simply 0 ?

Is it possible?

Let’s say I have a parametric surface S(u,v). Currently what I can do is create an arbitrary resolution quad mesh for this surface using the parametric surface function to determine the vertex locations and normals. When passed to a fragment shader, all vertex data is linearly interpolated which creates the linear faceted quad mesh that we see. If the quad mesh resolution is fine enough, you can’t really tell, but if I have a parametric representation of the surface then I shouldn’t have to settle for a linear approximation to my smooth surface!

If i could define my own custom interpolation scheme of the vertex data when things enter the fragment program of the shader then I can just use the parametric surface functions instead of the default linear interpolation. This would be give me a pixel perfect smooth surface that is consistent with the parametric description. However.. i can’t figure out if this is even possible.

Basically.. i want to control the data such as location of the stuff in between the vertices of the mesh, so that each face of the surface is pixel perfect smooth.

Another method I’ve thought of is using a signed distance field / ray matching approach. It would definitely work but it won’t really fit into my workflow so I don’t want to consider this option.

Any ideas?

———-

Edit: I have a feeling a geometry shader could be of use here? Currently reading about then but don’t really know anything about them!

I'm not a native English speaker, but I'll try my best getting the terminology right.. also, kind of a longshot for finding someone who can help me with this problem

I have a surface S with:

S := {(x,y,z) € R3 | z = 1/2y2, 0 <= x <= 2, -1 <= y <= 1}

I parametrized it, so that S = {x(x,y) | (x,y) € B} ==> x(x,y) = (x, y, 1/2y^(2))

For the surface integral I need the normal vector n(x,y) = x_x(x,y) x x_y(x,y),

which leads to: n(x,y) = (0, -y, 1)

That I got right, according to the short solutions that I have.

Now, the surface integral is where I struggle: (I add an image cause the integrals are too messy to write them here)

Calculation on Paper: Link to Image

Either I'm missing some identity of arsinh, for which 2*arsinh(1) = arsinh(3) or I'm messing up my integration but I don't see the error..

Edit: Found one mistake: The ln(...) of the solution equals the arcosh(3) not the arsinh(3). This confuses me even more tbh but hey..

https://imgur.com/a/4Zs6MUs

so far, I have managed to recreate the rose with x(u,v) = 1 + sin(8π u) sin(2π u) and y(u,v) = -1 + sin(8π u) cos(2π u), I am not very sure how to define the rotation, may I have some assistance

Question : https://imgur.com/a/4Zs6MUs

My thought process :
First, I want a 2D representation of the rose, so plugging in the values into the formula :

x(u,v) = 0

y(u,v) = 1 + sin(8π u) sin(2π u)

z(u,v) = 1 + sin(8π u) cos(2π u)

I am not very sure how do I manipulate 3/2π and -2 in this scenario, I think I have to first find a way to create a full loop of the rose then manipulate the height, but I do not have any idea on how to start, may I have some advice please

Hey, let's say I want to build a simple visualization tool for a calculus-in-multiple-variables course. I want to visualize character strings using procedurally generated surfaces. Specifically, I want to take (1) some input, (2) use that input to compute some specially defined functions describing the (x,y,z) coordinates of points on the surface as functions of two dummy parameters (u,v), and (3) and generate a simple static 3d rendering of the surface defined by the functions x(u,v), y(u,v), and z(u,v) for u and v chosen from some parameter range. I know how to do (1) and (2) already... But I am new to visualizations so I don't quite know where to start on (3).

I am aware of a huge number of tools that do this (Matlab does plotting, Wolfram Alpha does too, etc) but I want to try to do this in threejs to challenge myself, learn how threejs handles nuts and bolts of math I teach, and to have a convenient cool site to show off.

Anyone have recommended resources to learn how to solve this?

Hi /r/solidworks -- here's the existing .obj, derived from photogrammetry. i don't have the tools or skills to recreate it as a parametric surface. i'd like to use Onshape to extrude, cut, perforate and create custom versions

i can provide 4 forms of reference: physical sculpt (the most pleasing shape), photogrammetry photoset, raw photogrammetry, and clean polygonal model in .obj format (subtleties lost)

i'd prefer to work with someone in LA, because i could deliver the sculpt as opposed to waiting on Fedex

if you're interested, please PM me with your rate and availability, and also please let me see a similar / relevant model you created parametrically using Solidworks =:-)

on my end i'm happy to provide references re: what I'm like to work with

Hi.

So I have the parametric equation for the surface, its partial derivatives and normal vector. I want to move an object fitting onto it. I have a function that moves the objects, then the next is the draw. I get the movement's direction as partial derivating u and v, and summing these two. Then the surface's normal vector's these two's cross. Then I get the j orientation as k x i. My problem it doesn't give an appropriate orientation, the object is tilted a bit towards its side. I guess the the parital derivative should be looked at, by I calculated them 3 times, and I didn't find a mistake in it. Any ideas on it?

		ParamSurface * kleinSurface = new Klein();
		VertexData vd = kleinSurface->GenVertexData(pos.x, pos.y);

		vec3 i = normalize(vd.drdU + vd.drdV);
		vec3 k = normalize(vd.normal);
		vec3 j = normalize(cross(k, i));

		mat4 RotateTranslate = mat4(i.x, i.y, i.z, 0,
									j.x, j.y, j.z, 0,
									k.x, k.y, k.z, 0,
									vd.position.x, vd.position.y, vd.position.z, 1);

		mat4 RotateTranslateInv = mat4(	i.x, j.x, k.x, 0,
										i.y, j.y, k.y, 0,
										i.z, j.z, k.z, 0,
										-vd.position.x, - vd.position.y,   -vd.position.z, 1);



float U = u * 2.0f * (float)M_PI, V = v * 2.0f * (float)M_PI;
		float a = 6.0f * cosf(U) * (1+sinf(U));
		float b = 16.0f * sinf(U);
		float c = 4.0f * (1.0f - cosf(U) / 2.0f);

		if ((float)M_PI < U  && U <= (float) M_PI * 2.0f) {
			fx = a + c * cosf(V + (float)M_PI);
			fy = b;
			fz = c * sinf(V);

			vec3 drdU=vec3(6.0f * cosf(U)*cosf(U)-2.0f*sinf(U)*(3.0f*sinf(U)+cosf(V)+3.0f),
						16.0f*cosf(U), 
						2.0f*sinf(U)*sinf(V));

			vec3 drdV=vec3(-2*(cosf(U)-2.0f)*sinf(V),
						  0,
						  -2.0f*(cosf(U)-2)*cosf(V));

			vd.normal = cross(drdU, drdV);
			vd.position = vec3(fx, fy, fz);
			vd.drdU = drdU;
			vd.drdV = drdV;
		}

		else {
			fx = a + c * cosf(U) * cosf(V);
			fy = b + c * sinf(U) * cosf(V);
			fz = c * sinf(V);

			vec3 drdU= vec3(4*sinf(U)*cosf(U)*cosf(V)-2*sinf(U)*(3*sinf(U)+2*cosf(V)+3)+6*cosf(U)*cosf(U),
				-2*cosf(U)*cosf(U)*cosf(V)+4*cosf(U)*(cosf(V)+4.0f)+2.0f*sinf(U)*sinf(U)*cosf(V),
				2.0f * sinf(U)*sinf(V));

			vec3 drdV= vec3(2*(cosf(U)-2)*cosf(U)*sinf(V),
						2.0f*sinf(U)*(cosf(U)-2)*sinf(V),
						4.0f * (1.0f - cosf(U) / 2.0f)*cosf