Slate circular cone

At the right circular cone , the tip is perpendicular to the center of the base circle with the radius , at the oblique circular cone of (see drawing) perpendicular to a different point of the circular plane . The distance ( "error") of the point is from the center of the circle . The height is the distance between and the circular plane (with a straight circular cone the distance , with an inclined circular cone the distance ). The mantle of a circular cone is its surface without the area of ​​the base circle. For the benefit of a comfortable way of speaking, the route and route length, jacket and jacket surface, etc. are identified. Circular cones of the same height above the same base circle have the same volume (this follows from the Cavalier principle ). The envelope of the straight circular cone can be calculated elementarily:, where . There is no such simple formula for the mantle of the oblique circular cone. ${\displaystyle S}$ ${\displaystyle O}$ ${\displaystyle r}$ ${\displaystyle O}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle e}$ ${\displaystyle h}$ ${\displaystyle S}$ ${\displaystyle {\overline {SO}}}$ ${\displaystyle {\overline {SE}}}$ ${\displaystyle M}$ ${\displaystyle M=\pi rs}$ ${\displaystyle s^{2}=r^{2}+h^{2}}$

Cone shell

oblique circular cone

The jacket of the oblique circular cone is obtained through integration . The infinitesimal triangle (highlighted in green in the drawing) has the base

${\displaystyle r\cdot \mathrm {d} x,}$

where the angle is in the center of the circle, and the height (red line in the drawing) ${\displaystyle x}$

${\displaystyle f(x):={\sqrt {(r-e\cdot \cos(x))^{2}+h^{2}}},}$

hence the area

${\displaystyle {\frac {r}{2}}\cdot f(x)\cdot \mathrm {d} x.}$

Since symmetrically around is, it is sufficient from 0 to integrate, and double the result. Hence the formula for the mantle of the oblique circular cone is ${\displaystyle f(x)}$ ${\displaystyle \pi }$ ${\displaystyle \pi }$ ${\displaystyle M}$

${\displaystyle M=r\int \limits _{0}^{\pi }{\sqrt {(r-e\cdot \cos(x))^{2}+h^{2}}}\mathrm {d} x}$.

You can replace the minus in the integrand with a plus without changing the value of the integral (if you integrate over the full circle, you can even swap the cosine for the sine). The expression takes the straight circular cone into account as a special case and provides for the well-known formula. With a suitable choice of an upper and lower bound for the integrand, the jacket can be estimated upwards and downwards. From the estimate of the interval it follows, if it remains below a fixed limit, ${\displaystyle e=0}$ ${\displaystyle r+h}$

${\displaystyle {\frac {M}{2re}}\rightarrow 1}$for . ${\displaystyle e\rightarrow \infty }$

So if the deviation is large compared to the radius and height, the following applies approximately (diameter times deviation). Example: The mantle of the inclined circular cone with a radius of 2 cm, a height of 6 cm and a deviation of 50 cm has an area of ​​205.92 ... cm ² , i.e. about 2 times 2 times 50 cm ² (if the height would only be half as large, this would result in a coat of 201.85 ... cm ² ). ${\displaystyle M\approx 2re}$

Extreme value sentences

Of all circular cones of the same height above the same base circle, the one that has the smallest jacket (and thus the smallest surface).

Because if you take the mantle and the integrand as functions of , then if so ${\displaystyle e}$

${\displaystyle M(e)=r\int \limits _{0}^{\pi }f(e,x)\mathrm {d} x,}$

then

${\displaystyle M^{\prime }(e)=r\int \limits _{0}^{\pi }{\frac {\partial f(e,x)}{\partial e}}\mathrm {d} x.}$

After the partial derivation of the integrand one recognizes that for and otherwise (in addition ). is therefore monotonically increasing (the jacket becomes larger with increasing deviation). The line is called the axis of the circular cone. If the axis maintains a constant length, i.e. if ( fixed) and thus , then applies again to , but is now , has a maximum. Therefore the following applies: ${\displaystyle f(e,x)}$ ${\displaystyle M'(e)=0}$ ${\displaystyle e=0}$ ${\displaystyle M'(e)>0}$ ${\displaystyle M''(0)>0}$ ${\displaystyle M(e)}$ ${\displaystyle {\overline {SO}}}$ ${\displaystyle e^{2}+h^{2}=l^{2}}$ ${\displaystyle l}$ ${\displaystyle h^{2}=l^{2}-e^{2}}$ ${\displaystyle M'(e)=0}$ ${\displaystyle e=0}$ ${\displaystyle M''(0)<0}$ ${\displaystyle M(e)}$ ${\displaystyle e=0}$

Of all circular cones of the same axis over the same base circle, the one that has the largest envelope (and thus the largest surface).

Relationship between a straight ellipse and an oblique circular cone

Section through a straight elliptical cone

A straight elliptical cone with the height and the ellipse ${\displaystyle h}$

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$

as the base ( ) is described by the following equation: ${\displaystyle a>b}$

${\displaystyle F(x,y,z):={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{h^{2}}}=0.}$

All values that meet ( ) lie on the surface of the straight elliptical cone, the tip of which is the zero point (see drawing). be the minimum angle of repose ( ) and the maximum ( ), so . A plane whose normal lies in the yz plane and which forms the angle with the xy plane , where ${\displaystyle (x,y,z)}$ ${\displaystyle F(x,y,z)=0}$ ${\displaystyle 0\leq z\leq h}$ ${\displaystyle \alpha }$ ${\displaystyle \tan \alpha =h/a}$ ${\displaystyle \beta }$ ${\displaystyle \tan \beta =h/b}$ ${\displaystyle \alpha <\beta }$ ${\displaystyle \gamma }$

${\displaystyle \cos \gamma ={\frac {\cos \beta }{\cos \alpha }}}$

intersects the straight elliptical cone in a circle. Its radius is ${\displaystyle r}$

${\displaystyle r=n\cdot {\frac {a^{2}}{bh}}}$

here means the distance between the cutting plane and the apex of the cone (perpendicular to the plane), if necessary you have to think of the cone as being extended beyond the base in order to be able to carry out the complete circular section. The remaining cone between the tip and the cut surface (indicated in yellow in the drawing) is therefore an oblique circular cone. Every straight elliptical cone contains an oblique circular cone and vice versa: Every oblique circular cone can be expanded to a straight elliptical cone. ${\displaystyle n}$

See also

• Oblique elliptical cone
• Slate cone (general)