area of spesific shape

[niksirat]

This question may be simple, but I could not calculate in principle and accurately.

The equation of the cone is [imath]z=\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}[/imath]. My goal is to calculate a part of the surface of the cone, which is located between two planes [imath]z=z_0[/imath] and [imath]z=z_1 > z_0[/imath] so that the interface between plane [imath]z=z_0[/imath] and cone is an ellipse with large radius a and small radius b and the distance between the top plane and the bottom plane is [imath]h_0[/imath].

 

[Deleted member 4993]

This question may be simple, but I could not calculate in principle and accurately.

The equation of the cone is [imath]z=\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}[/imath]. My goal is to calculate a part of the surface of the cone, which is located between two planes [imath]z=z_0[/imath] and [imath]z=z_1 > z_0[/imath] so that the interface between plane [imath]z=z_0[/imath] and cone is an ellipse with large radius a and small radius b and the distance between the top plane and the bottom plane is [imath]h_0[/imath].

Do you know the process to calculate the surface area of the WHOLE come via integration?

 

[niksirat]

Do you know the process to calculate the surface area of the WHOLE come via integration?

I think surface area of [imath]z[/imath] when projection of [imath]z=f(x,y)[/imath] on [imath]xy[/imath] plane is [imath]D[/imath] calculate with this formula:
[math]\iint_D\sqrt{1+z_x^2+z_y^2} ~~dx~dy.[/math]My purpose of asking this question is to calculate the RED area of the figure below, whose base is an ellipse with a large diameter [imath]W[/imath] and a small diameter [imath]L[/imath]:

 

[blamocur]

I think your best bet is numeric computation. I wrote a quick-and-dirty (my favorite MO) for the area of the whole cone, i.e. [imath]z_0 = 0[/imath] script which has a surprisingly high precision if the integration steps are small enough. It passes a simple test (comparison with regular cone, i.e. when [imath]a=b[/imath]), but it still might have bugs, so I am posting it as a spoiler in case someone might find bugs in there. Also, it can be improved using better numerical integration, but I am too lazy for that.

Spoiler

import numpy as np

def ds (u,v, a, b):
    a2  = a**2;
    b2  = b**2;
    u2  = u**2;
    sv2 = np.sin (v)**2;
    cv2 = np.cos (v)**2;
    return np.sqrt (u2 * (a2 * b2 + a2 * sv2 + b2 * cv2));

def ellipticConeSurface (a, b, h, nn=1000):
    u1    = np.linspace (0, h, nn);
    v1    = np.linspace (0, 2*np.pi, nn);
    duv   = h * 2 * np.pi / (nn**2);
    u2,v2 = np.meshgrid (u1, v1);
    area    = duv * np.sum (ds (u2, v2, a, b));
    return area;

def utEllipticConeSurface ():
    R = 3;
    H = 5;
    s1 = ellipticConeSurface (R/H, R/H, H);
    s2 = np.pi * R * np.sqrt (H**2 + R**2);
    print (s1, " = ", s2, " = ", s1-s2);

utEllipticConeSurface ();

 

[Dr.Peterson]

I think surface area of [imath]z[/imath] when projection of [imath]z=f(x,y)[/imath] on [imath]xy[/imath] plane is [imath]D[/imath] calculate with this formula:
[math]\iint_D\sqrt{1+z_x^2+z_y^2} ~~dx~dy.[/math]My purpose of asking this question is to calculate the RED area of the figure below, whose base is an ellipse with a large diameter [imath]W[/imath] and a small diameter [imath]L[/imath]:
View attachment 34283

You haven't clearly shown us what you tried, and where you got stuck; are you aware of the fact that the perimeter of the ellipse can't be calculated in closed form, and the same is true of some facts about elliptical cones? (See here, for example.) A frustum of an elliptical cone won't be any easier!

I found this site that uses an approximate formula I can't vouch for:

Truncated Elliptic Cone - Geometry Calculator

Calculations of geometric shapes and solids: Truncated Elliptic Cone.

rechneronline.de

 

[Steven G]

Dr Khan,
Once gain I am confused. To be a cone, shouldn't a=b in the formula for Z in the op??? Thanks for your time!

 

[Dr.Peterson]

Dr Khan,
Once gain I am confused. To be a cone, shouldn't a=b in the formula for Z in the op??? Thanks for your time!

The word cone is often used specifically of a right circular cone, but it also has a far more general meaning, which often has to be determined from context (which here is the given equation and the explicit mention of an ellipse). See

Cone -- from Wolfram MathWorld

A (finite, circular) conical surface is a ruled surface created by fixing one end of a line segment at a point (known as the vertex or apex of the cone) and sweeping the other around the circumference of a fixed circle (known as the base). When the vertex lies above the center of the base (i.e...

mathworld.wolfram.com

Quoting from there:

As can be seen from the above, care is needed when interpreting the unqualified term "cone" since, depending on context, it may refer to the right or oblique configurations, circular or elliptical bases, the single- or double-napped versions, the finite or infinite surface excluding the circular/elliptical base, the finite surface including it, or the finite solid bounded by the sides and base. When used without qualification, especially in elementary contexts, the term "cone" often means the filled (solid) right circular cone.​

In fact, it can go beyond that, allowing all sorts of bases:

Generalized Cone -- from Wolfram MathWorld

A ruled surface is called a generalized cone if it can be parameterized by x(u,v)=p+vy(u), where p is a fixed point which can be regarded as the vertex of the cone. A generalized cone is a regular surface wherever vy×y^'!=0. The above surface is a generalized cone over a cardioid. A generalized...

mathworld.wolfram.com

So there is nothing wrong with calling this elliptical cone a cone, given that we have sufficient information about it. You can even call a pyramid a cone, if you want to.