Oblique elliptical cone

oblique elliptical cone

The oblique elliptical cone (English: oblique cone) is a generalization of the oblique circular cone ; its base is an ellipse with corresponding semiaxes and . The tip of the oblique cone does not need to be above the center of the ellipse , but can be above . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle S}$ ${\displaystyle O}$ ${\displaystyle S_{0}=(u,v)}$

Floor space

The base is formed by an ellipse :

${\displaystyle A=\pi ab=\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}}$

With the length of the major and minor semi-axes and ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \varepsilon ={\frac {\sqrt {a^{2}-b^{2}}}{a}}\in [0,1)}$

volume

The generalized formula of the inclined circular cone applies to the volume :

${\displaystyle V={\frac {1}{3}}hA={\frac {1}{3}}h\pi ab={\frac {\pi }{3}}hab}$

with as the height of the crooked cone, ${\displaystyle h}$

${\displaystyle a={\frac {d_{\mathrm {max} }}{2}}}$   

as the length of the large (half maximum diameter) and

${\displaystyle b={\frac {d_{\mathrm {min} }}{2}}}$   

of the small semiaxes (half the minimum diameter).

Rule of thumb

${\displaystyle 1{,}0498\cdot hab\geq V={\frac {\pi }{3}}hab\geq 1{,}0497\cdot hab\geq hab}$

The error when using to calculate the volume is therefore less than 5% (factor 1.05) and can be neglected in an estimate. ${\displaystyle hab}$

In general: mantle of the oblique elliptical cone

oblique elliptical cone

The calculation of the surface area is demanding.

The ellipse is through

${\displaystyle x(t)=a\cdot \cos(t)}$

${\displaystyle y(t)=b\cdot \sin(t)}$

described ( from , parameter representation, see drawing). ${\displaystyle t}$ ${\displaystyle [0,2\pi ]}$

Be it

${\displaystyle N(t):={\sqrt {a^{2}\cdot \sin(t)^{2}+b^{2}\cdot \cos(t)^{2}}}}$

The base of the infinitesimal triangle (which is used to calculate the envelope of the cone) is

${\displaystyle dU=N(t)\cdot \mathrm {d} t}$

this follows through differentiation from the parameter representation above. Often referred to in the literature as ${\displaystyle N(t)}$

${\displaystyle N(t)=a\cdot {\sqrt {1-n^{2}\cdot \cos(t)^{2}}}}$

written. with means "numerical eccentricity". The integration of to results in an "elliptical integral of the second kind" (this is the well-known formula for the circumference of an ellipse). The infinitesimal triangle lies in the plane through the ellipse tangent ${\displaystyle n}$ ${\displaystyle n^{2}=(a^{2}-b^{2})/a^{2}}$ ${\displaystyle t=0}$ ${\displaystyle 2\pi }$

${\displaystyle P=(a\cdot \cos(t),b\cdot \sin(t))}$

and is defined by the cone apex at a distance vertically above . The height of the infinitesimal triangle is ${\displaystyle S}$ ${\displaystyle h}$ ${\displaystyle E=(u,v)}$

${\displaystyle f(t)={\sqrt {k(t)^{2}+h^{2}}}}$

(not to be confused with the height of the cone). Here the perpendicular from means to the ellipse tangent to the point . Be it ${\displaystyle h}$ ${\displaystyle k(t)}$ ${\displaystyle E}$ ${\displaystyle P}$

${\displaystyle Z(t):=ab-va\cdot \sin(t)-ub\cdot \cos(t)}$

Then applies

${\displaystyle k(t)={\frac {Z(t)}{N(t)}}}$

The surface of the infinitesimal triangle is thus

${\displaystyle {\frac {1}{2}}\cdot f(t)\cdot N(t)\cdot dt={\frac {1}{2}}{\sqrt {Z(t)^{2}+h^{2}\cdot N(t)^{2}}}\mathrm {d} t}$

The formula for the lateral surface M of the oblique elliptical cone is therefore:

${\displaystyle M={\frac {1}{2}}\int \limits _{0}^{2\pi }{\sqrt {Z(t)^{2}+h^{2}\cdot N(t)^{2}}}\mathrm {d} t}$

Since the integrand is not symmetrical , you have to integrate over the full circle. Under the integral from 0 to one can replace the minus signs in together with plus signs. Then the formula is written out ${\displaystyle \pi }$ ${\displaystyle 2\pi }$ ${\displaystyle Z(t)}$

${\displaystyle M={\frac {1}{2}}\int \limits _{0}^{2\pi }{\sqrt {(ab+va\cdot \sin(t)+ub\cdot \cos(t))^{2}+h^{2}\cdot (a^{2}\cdot \sin(t)^{2}+b^{2}\cdot \cos(t)^{2})}}\mathrm {d} t}$

Instead of and , one can also choose and as integration limits without changing the value. If one understands as a function of and , then it serves as generator of the known formulas for circle, ellipse and cone. ${\displaystyle 0}$ ${\displaystyle 2\pi }$ ${\displaystyle -\pi }$ ${\displaystyle \pi }$ ${\displaystyle M}$ ${\displaystyle a,b,u,v}$ ${\displaystyle h}$

${\displaystyle M(r,r,0,0,0)}$ = Circular area

${\displaystyle M(a,b,0,0,0)}$ = Elliptical surface

${\displaystyle M(r,r,0,0,h)}$ = Outer surface of the straight circular cone

${\displaystyle M(r,r,e,0,h)=M(r,r,0,e,h)}$ = Outer surface of the oblique circular cone

${\displaystyle M(a,b,0,0,h)}$ = Lateral surface of the straight elliptical cone

${\displaystyle M(a,b,u,v,h)}$ = Outer surface of the oblique elliptical cone.

An extreme value theorem

If you move the tip of the oblique elliptical cone at a constant height (or with a constant axis) over the beam ( c any slope), then the jacket is a differentiable function of (with a function of v ). The following applies and (or ) and thus the sentence (analogous to the circular cone) ${\displaystyle S}$ ${\displaystyle v(u)=cu}$ ${\displaystyle u}$ ${\displaystyle u=0}$ ${\displaystyle M'(0)=0}$ ${\displaystyle M''(0)>0}$ ${\displaystyle M''(0)<0}$

Among all elliptical cones of the same height (the same axis) above the same basic ellipse, the one that has the smallest (or largest) jacket.

In the proof using the fact that the differentiation according to drag under the integral can and that the following integrand integrated over the full circle, disappear: , and wherein denotes a function that order is symmetrical, z. B. or . ${\displaystyle u}$ ${\displaystyle G(t)\sin(t)}$ ${\displaystyle G(t)\sin(2t)}$ ${\displaystyle G(t)\cos(t)}$ ${\displaystyle G(t)}$ ${\displaystyle {\frac {i}{2}}\pi \ (i=1,2,3)}$ ${\displaystyle G(t)=\cos(2t)}$ ${\displaystyle G(t)=a^{2}\sin(t)^{2}+b^{2}\cos(t)^{2}}$

Special: jacket of the straight elliptical cone

For (i.e. for the straight elliptical cone) the mantle formula is ${\displaystyle u=v=0}$

${\displaystyle M={\frac {1}{2}}\int \limits _{0}^{2\pi }{\sqrt {a^{2}b^{2}+h^{2}\cdot (a^{2}\cdot \sin(t)^{2}+b^{2}\cdot \cos(t)^{2})}}\mathrm {d} t}$

By the allowed trick

${\displaystyle a^{2}b^{2}=a^{2}b^{2}\cdot (\sin(t)^{2}+\cos(t)^{2})}$

the integrand can be ordered according to and , and one obtains the expression ${\displaystyle \sin(t)^{2}}$ ${\displaystyle \cos(t)^{2}}$

${\displaystyle M={\frac {1}{2}}\int \limits _{0}^{2\pi }{\sqrt {A^{2}\cdot \sin(t)^{2}+B^{2}\cdot \cos(t)^{2}}}\mathrm {d} t}$

where and . The integral (without the factor ½) means the circumference of the ellipse with the semiaxes and . Therefore the following applies: ${\displaystyle A^{2}:=a^{2}(b^{2}+h^{2})}$ ${\displaystyle B^{2}:=b^{2}(a^{2}+h^{2})}$ ${\displaystyle A}$ ${\displaystyle B}$

The lateral surface of the straight elliptical cone with the semi-axes and and the height is numerically equal to half the circumference of the ellipse with the semi-axes and ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle h}$ ${\displaystyle A}$ ${\displaystyle B}$

The benefit of this theorem is that the known estimates for the circumference of the ellipse can now be applied to the mantle calculation. For the circumference of the ellipse with the semi-axes and the following applies in a first approximation ( and , therefore also ) ${\displaystyle U}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle a\neq b}$ ${\displaystyle h>0}$ ${\displaystyle A\neq B}$

${\displaystyle U>\pi \cdot (A+B)}$

The estimate for the jacket of the straight elliptical cone is obtained from this ${\displaystyle M}$

${\displaystyle M>{\frac {1}{2}}\cdot \pi \cdot \left(a\cdot {\sqrt {b^{2}+h^{2}}}+b\cdot {\sqrt {a^{2}+h^{2}}}\right)}$

The equals sign applies to (envelope of the straight circular cone) or (elliptical or circular area). For example , and . The estimate gives the value 36.7 ... The exact value is 36.9 ... ${\displaystyle a=b}$ ${\displaystyle h=0}$ ${\displaystyle a=3}$ ${\displaystyle b=2}$ ${\displaystyle h=4}$

Final remark: By estimating the integrand upwards and downwards one obtains the rough inequality for (the equal sign applies to or ). The surface area is therefore approximately equal to the arithmetic mean of the lower and upper bound. ${\displaystyle \pi B<M<\pi A}$ ${\displaystyle b<a}$ ${\displaystyle a=b}$ ${\displaystyle h=0}$

See also

• Slate circular cone
• Slate cone (general)