Talk:Paraboloid

Paraboloid value proof

First need to divide paraboloid into 10 parts, which have equal height $h=0.1$. Each part is cylinder with height $h=0.1$. It is so, that paraboloid is divided on z axis, without touching any over axis (y or x). So z axis is divided with step 0.1 from 0 to 1, like this: 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1. So radius of each of 10 cylinders depending on certain z position, but paraboloid reminds us parabola, so we can take paraboloid projection onto zOy plane/surface and will see parabola $z=y^{2}$. So now radius of each cylinder depending what is $z_{1}=0.1$, $z_{2}=0.2$, $z_{3}=0.3$, ..., $z_{10}=1$. And radius of each cylinder is $r={\sqrt {z}}$, because say at point $z=0.1$ radius (of smallest cylinder) is $r_{1}=y_{1}={\sqrt {z_{1}}}={\sqrt {0.1}}\approx 0.316$. So $r_{2}={\sqrt {0.2}}$; $r_{3}={\sqrt {0.3}}$; $r_{4}={\sqrt {0.4}}$; $r_{5}={\sqrt {0.5}}$; $r_{6}={\sqrt {0.6}}$; $r_{7}={\sqrt {0.7}}$; $r_{8}=0.8^{0.5}$; $r_{9}=0.9^{1/2}$; $r_{10}={\sqrt {1}}=1.$ And cylinder value formula is $V_{c}=\pi r^{2}h$. So $h=0.1$ for each cylinder and so each cylinder values are those:

$V_{1}=r_{1}^{2}\pi h=({\sqrt {0.1}})^{2}\pi h=0.1\pi h=0.1\pi \cdot 0.1=0.01\pi ;$

$V_{2}=r_{2}^{2}\pi h=({\sqrt {0.2}})^{2}\pi h=0.2\pi 0.1=0.02\pi ;$

$V_{3}=r_{3}^{2}\pi h=0.3\pi h=0.03\pi$; $V_{4}=0.3\pi h;$ $V_{5}=0.5\pi h$; $V_{6}=0.6\pi h;$ $V_{7}=0.7\pi h$; $V_{8}=0.8\pi h;$ $V_{9}=0.9\pi h,$ $V_{10}=\pi h.$

So paraboloid value up to $z=1$ is:

$V=\pi h(r_{1}+r_{2}+r_{3}+r_{4}+r_{5}+r_{6}+r_{7}+r_{8}+r_{9}+r_{10})=0.1\pi (0.1+0.2+0.3+0.4+0.5+0.6+0.7+0.8+0.9+1)=$

$=0.1\pi \cdot 5.5=0.55\pi \approx 1.727876.$

Precise (calculating with integrals method) paraboloid value of function $z=x^{2}+y^{2}$ up to $z=1$ is $V={\pi \over 2}=0.5\pi =1.5708$

Dividing into more parts/cilinders you can get paraboloid value arbitrary precise. —Preceding unsigned comment added by 212.59.24.214 (talk) 14:07, 11 February 2010 (UTC)

This thing $V=h(r_{1}+r_{2}+r_{3}+r_{4}+r_{5}+r_{6}+r_{7}+r_{8}+r_{9}+r_{10})=0.1(0.1+0.2+0.3+0.4+0.5+0.6+0.7+0.8+0.9+1)$ can be puted into integral, $r_{max}=z_{max}=z=1$ as $\int _{0}^{1}zdz={z^{2} \over 2}|_{0}^{1}={1^{2} \over 2}-{0^{2} \over 2}={1 \over 2}$ and this thing need just multiply by $\pi$ and then you can easily calculate paraboloid value if just know height z. $V=0.5\pi$.

For example if z=100, then if x=0, then y=10, because $z=y^{2}=10^{2}=100$ and paraboloid formula $z=x^{2}+y^{2}$ is like parabola $z=y^{2}$, only assume x=0. So then calculation is very simple:

$V=\pi \int _{0}^{100}zdz=\pi {z^{2} \over 2}|_{0}^{100}=\pi ({100^{2} \over 2}-{0^{2} \over 2})=5000\pi .$ This very logical replacing next method of finding paraboloid value if z=100, $\phi =0$ and up to $\phi =2\pi$; $z=x^{2}+y^{2}$ <=> $z=\rho ^{2}$, because $x^{2}+y^{2}=\rho ^{2}$ in Cylindrical coordinate system, $\rho$ is also maximum possible radius of paraboloid, so value of paraboloid is:

$V=\iiint _{V}dxdydz=\iiint _{T}\rho d\rho d\phi dz=\int _{0}^{2\pi }d\phi \int _{0}^{10}\rho d\rho \int _{\rho ^{2}}^{100}dz=$

$=\int _{0}^{2\pi }d\phi \int _{0}^{10}\rho (100-\rho ^{2})d\rho =\int _{0}^{2\pi }({100\rho ^{2} \over 2}-{\rho ^{4} \over 4})|_{0}^{10}d\phi =\int _{0}^{2\pi }(50\cdot 10^{2}-{10^{4} \over 4})d\phi =$ $=\int _{0}^{2\pi }(5000-2500)d\phi =2500\int _{0}^{2\pi }d\phi =5000\pi .$

And if you going to calculate paraboloid $z=x^{2}+y^{2}$ value if z=9, then $z=\rho ^{2}=9$, $\rho =3$; then you have to do all this cylindrical coordinates hell:

$V=\iiint _{V}dxdydz=\iiint _{T}\rho d\rho d\phi dz=\int _{0}^{2\pi }d\phi \int _{0}^{3}\rho d\rho \int _{\rho ^{2}}^{9}dz=$

$=\int _{0}^{2\pi }d\phi \int _{0}^{3}\rho (9-\rho ^{2})d\rho =\int _{0}^{2\pi }({9\rho ^{2} \over 2}-{\rho ^{4} \over 4})|_{0}^{3}d\phi =\int _{0}^{2\pi }({9\cdot 3^{2} \over 2}-{3^{4} \over 4})d\phi =$ $=\int _{0}^{2\pi }({81 \over 2}-{81 \over 4})d\phi =\int _{0}^{2\pi }{162-81 \over 4}d\phi ={81 \over 4}\int _{0}^{2\pi }d\phi ={81\pi \over 2}.$

And with obvious method (because of this my proof) to calculate paraboloid $z=x^{2}+y^{2}$ value it will takes much less time. So z=9 and paraboloid value is:

$V=\pi \int _{0}^{9}z\;dz=\pi {z^{2} \over 2}|_{0}^{9}=\pi ({9^{2} \over 2}-{0^{2} \over 2})={81\pi \over 2}.$ —Preceding unsigned comment added by 212.59.24.214 (talk) 18:33, 11 February 2010 (UTC)

Here paraboloid value proof again if each cylinder height h=1 and paraboloid divided into 100 such cylinders; each cylinder value can be calculated $\pi r_{n}^{2}h$. So on z axis there steps by 1, like $z_{1}=1$, $z_{2}=2$, $z_{3}=3$,..., $z_{99}$=99, $z_{100}=100$. We projecting paraboloid onto yOz plane, so then $z_{n}=y_{n}^{2}=r_{n}^{2}$, so $y_{n}$ is radius and $y_{n}={\sqrt {z_{n}}}=r_{n}$. So total paraboloid value is:

$V=V_{1}+V_{2}+V_{3}+...+V_{99}+V_{100}=\pi r_{1}^{2}h+h\pi r_{2}^{2}+h\pi r_{3}^{2}+h\pi r_{4}^{2}+...+h\pi r_{99}^{2}+h\pi r_{100}^{2}=$

$=h\pi (({\sqrt {1}})^{2}+({\sqrt {2}})^{2}+({\sqrt {3}})^{2}+({\sqrt {4}})^{2}+...+({\sqrt {99}})^{2}+({\sqrt {100}})^{2})=$ $=1\cdot \pi (1+2+3+4+...+98+99+100)=5050\pi .$

And for comparision if to divide only into 10 parts and if h=10 then, z=100 still, so then such answer: $V=10\cdot \pi (10+20+30+40+50+60+70+80+90+100)=5500\pi .$

You can imagine this sumations like dividing place under line y=x into peaces of width a=1 and height $b_{n}$, $b_{1}=1$, $b_{2}=2,$ $b_{3}=3$, ... , $b_{99}=99$, $b_{100}=100$. If you multiply each column height $b_{n}$ with width a=1, then you will get column area. If you sum up all 100 columns (1*1+1*2+1*3+...+1*98+1*99+1*100=5050), then you will get (approximate) area under line y=x and this line from 0 to 100 makes triangle. And this triangle area $S={100^{2} \over 2}=5000.$ By integrating you geting the same $\int _{0}^{100}x\;dx={x^{2} \over 2}|_{0}^{100}={100^{2} \over 2}-{0^{2} \over 2}=5000$. And by multiplying with $\pi$ you getting paraboloid value $V=5000\pi .$ —Preceding unsigned comment added by 212.59.24.214 (talk) 19:56, 11 February 2010 (UTC)

shrimp are tasty

so in nature the ONLY place where a hyperbolic paraboloid is found is a shrimp? I have trouble with this..... —Preceding unsigned comment added by 75.46.90.105 (talk) 20:56, 7 May 2010 (UTC)

there have to be other examples of objects with paraboloid than the mantis shrimp mentioned in "in nature" — Preceding unsigned comment added by 75.46.90.105 (talk) 20:59, 7 May 2010 (UTC)

None of this makes sense to a non-Maths person

I came to this page because it was linked to from the Pringles page. The shape of those crisps was described as being a hyperbolic paraboloid. I wondered what this means, so I clicked on the link and... Uh... That's where the comprehensible part of my story finishes. This page has absolutely no meaning whatsoever to someone who doesn't have advanced knowledge of Mathmatics. At least, I think that the knowledge is Maths based - it seems that I don't even know what I'd need to know to understand the info on this page. Basically, not very useful to anyone wondering what shape a Pringle is...86.166.154.225 (talk) 01:21, 9 October 2011 (UTC)

Well, what kind of understanding of the shape did you hope to get without the math? —Tamfang (talk) 02:50, 31 August 2013 (UTC)

It might be worth removing the Pringles reference, since it's a commercial product, and this is a page about math. Keeping it could even give Pringles a bad name, since companies in the past (e.g. "The North Face") have used replaced wikipedia images as part of their advertising campaign. —Whole Oats (talk) 06:40, 28 June 2022 (UTC)

Coordinate System

The article states: "In a suitable coordinate system with three axes x, y, and z"

Thus, a reader comes to this article to try model a paraboloid and is left wondering what the coordinate system is; ie the article is effectively useless. — Preceding unsigned comment added by 212.183.140.43 (talk) 11:06, 30 October 2011 (UTC)

Added link to Coordinate system. Y

A suitable coordinate system is one that allows the mathematical operations required for the discussion. In this case, one must be able to use Addition, Negation, Multiplication, Division, and Exponentiation operations. An angular axes system, Polar or Spherical for instance, does not allow the operation of multiplication or division with the same Translational meaning as in a Cartesian coordinate system.
Should "Suitable coordinate system" be created as an article?
SBaker43 (talk) 12:42, 13 February 2014 (UTC)

Criteria for stackability

One of two "applications" mentioned in the article is this:

"The manufacturers of a widely-sold fried snack food, Pringles ersatz potato chips, utilize a hyperbolic paraboloid shape in order to produce a stackable snack."

But being a hyperbolic paraboloid has virtually nothing to do with stackability. Any surface for the form z = f(x,y), i.e., that is the graph of a function, is stackable. That is, its vertical translates z = f(x,y) + c where c ranges over an interval [a,b] are disjoint surfaces that fill the solid region between z = f(x,y) + a and z = f(x,y) +b.Daqu (talk) 22:55, 16 November 2011 (UTC)

discrimination

For a general paraboloid of the form $z=ax^{2}+bxy+cx^{2}+dx+ey+f$, am I right in guessing that it's hyperbolic if $b^{2}>4ac$, elliptic if $b^{2}<4ac$, and a cylinder (ruled parallel to the xy plane) if they're equal? —Tamfang (talk) 05:28, 31 August 2013 (UTC)

Splitting of hyperbolic paraboloid (may 2016)

I would suggest to split this article in two articles: paraboloid mostly about the eliptical paraboloid and the common features and a seperate article for the hyperbolic paraboloid.

My main reason is the paraboloids don't even look the same and by splitting both can improve. because the eliptical paraboloid is I think the most expected WP:Least I suggest to keep paraboloid for the eliptical one. WillemienH (talk) 11:08, 15 May 2016 (UTC)

I don't think there different looks are sufficient reason to split the article. In my experience in literature they are usually treated together and it seems that currently most/all other language Wikis treat them in one article. While Mathworld actually does have separate articles for them, the encyclopedia of mathematics and britannica do not.--Kmhkmh (talk) 21:24, 5 June 2016 (UTC)

@WillemienH:, I'm going to have to agree with @Kmhkmh:. I think It's nice to have them both in the same article, so that you can compare/contrast. If they were split apart, it might just be more confusing both for readers and editors. Popcrate (talk) 09:56, 22 February 2017 (UTC)

This article is short, almost a stub, and I see no good reason for splitting. When WillemienH thinks that elliptical paraboloids are the most expected, this is a unsourced personal opinion. He forgets that everybody can easily see hyperbolic paraboloids in everyday life (see the end of the article).

Instead of splitting, it would be better to explain why two apparently different shapes share the name of paraboloid. The fundamental reason is that a paraboloid is a quadric surface that is tangent to the plane at infinity. This should appear in the body of the article, but is to technical for the lead (or an explanation section), and I am looking for a more elementary explanation. D.Lazard (talk) 11:28, 22 February 2017 (UTC)

A special feature of both the paraboloids is: Obviously they contain a lot of parabolas. If the paraboloids are represented as in the lead, any plane, which is parallel to the z-axis, intersects a paraboloid in a parabola (with rare exceptions in the hyperbolic case). Parabolas appear on cones and hyperboloids of one sheet, too. But not obviously. --Ag2gaeh (talk) 16:05, 22 February 2017 (UTC)

Parabolas appear also on two-sheets hyperboloids. I knew this characterization, but it is difficult to state it properly. Maybe a correct formulation is : "a paraboloid is a quadric surface that is not a cylinder, and such there exists a line, called the axis, such that almost all planes, that are parallel to this direction, cut the surface into a parabola". IMO, this would be unclear for many readers, and would need a specific section for a detailed explanation. I find clearer (at least more constructive) the (almost equivalent) formulation that I have added to the lead, although it deserves also to be detailed in the body of the article. D.Lazard (talk) 16:33, 22 February 2017 (UTC)

Sorry, I did not intend to define a paraboloid by plane sections. I just wanted to motivate their names e/h paraboloid. Yes, I forgot to mention (above) the two sheeted hyperboloid.--Ag2gaeh (talk) 17:53, 22 February 2017 (UTC)

Use Simpler Linear Diameter equation

Under the section: Dimensions of a paraboloidal dish "A more complex calculation is needed to find the diameter of the dish measured along its surface." The equation shown is unnecessarily complex. You can define the paraboloid with two variables. Here they use three. Is there a reason for this? The final form isn't any cleaner, so that's not a reason. I would suggest getting rid of the Ps and Qs.

Derivation: To get the arc length along the inside of the paraboloid or "linear diameter", measure the length along the curve of the parabola. Using the arc length formula,

Diameter = s = Integrate[Sqrt[1+(dy/dx)^2],{x,a,b}]. 

Choosing a and b to be defined in terms of R, the rim radius of the top of the paraboloid, and knowing y=x^2/(4F) is the general formula for a parabola, where F is the focus, we get:

s = Integrate[Sqrt[1+x^2/(4F^2)],{x,-R,R}],

which is:

s = diameter = (1/2)R*Sqrt[4+R^2/F^2]+2F*ArcSinh[R/(2F)]

You can also substitute ArcSinh[x] for the natural log form if desired:

s = (1/2)R*Sqrt[4+R^2/F^2]+2F*Log[R/(2F)+Sqrt[(R/(2F))^2+1]]

This only uses two variables.

Note:I was plugging this straight into mathematica which is why I wrote the equations this way.

EvanN90 (talk) 01:45, 15 June 2018 (UTC)

z=x

I don't know where in Wikipedia to post this. Your z looks like your x. Please use a fond for your Maths that is easy to read. Mathematics on Wikipedia is brilliant. Where do I post glowing praise from a lonely autodidact? — Preceding unsigned comment added by Norm Tered (talk • contribs) 23:34, 30 October 2020 (UTC)

They do, indeed, look similar: $x,z.$ This is an artifact of the font. I suggest zooming to a higher magnification on the device you are using to read the articles.—Anita5192 (talk) 00:04, 31 October 2020 (UTC)

"Definition" is not a definition

The section Hyperbolic paraboloid contains this paragraph:

"These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed skew lines."

The problem is that this is not a definition, because many other surfaces also fit this description.

I hope that someone knowledgeable about this subject can fix this problem. 2601:200:C000:1A0:EC4E:3D24:A106:AEAE (talk) 18:31, 16 July 2021 (UTC)

"Many other surfaces also fit this description": No, this characterizes hyperbolic paraboloids. If you disagree, provide an example and a source. D.Lazard (talk) 20:31, 16 July 2021 (UTC)