Slate cone

Oblique cone

The basis of the general skew cone is a closed curve with the parameter representation x (t): = p (t) and y (t): = q (t), where p and q are differentiable in the interval [c, d] ( apart from a finite number of exceptions), also: p (c) = p (d) and q (c) = q (d). The point E = (u, v) lies in the plane of the curve, the cone tip S is at a distance h perpendicular to E, i.e. S = (u, v, h). The following formalism also applies to non-closed curves, in which case it is better to speak of sails than of cones (triangular sails, curved triangles). In order to keep the formulas clear, the derivation with respect to t is provided with a point (as is usual in physics).

Mantle of the general oblique cone

The formula for the lateral surface M of the general oblique cone is similar to that of the oblique elliptical cone (apart from the integration limits):

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

Mean here

${\displaystyle Z(t)=(p-u)\cdot {\dot {q}}-(q-v)\cdot {\dot {p}}}$

and

${\displaystyle N(t)={\sqrt {{\dot {p}}^{2}+{\dot {q}}^{2}}}}$

One could also calculate the pyramid jacket with this formalism (the pyramid as a "cone" with a square base), but here the elementary geometry leads more quickly to the goal.

The geometric meanings of Z and N

From Z

Extracting a function f over [c, d] requires care, because the square root of f² is ambiguous, even infinitely ambiguous. To see this, one only needs to reverse the value f (a) to -f (a) at any point a (which is not the zero point of f). The roots | f | are geometrically significant and f. If the height h tends towards zero in the formula for the mantle of a general oblique cone, the expression arises

${\displaystyle M={\frac {1}{2}}\int \limits _{c}^{d}{\sqrt {Z(t)^{2}}}\mathrm {d} t}$

and in particular for the root | Z (t) |:

${\displaystyle M={\frac {1}{2}}\int \limits _{c}^{d}|Z(t)|\mathrm {d} t}$

From a geometrical point of view, this is the area of ​​the "folded" cone jacket in the xy plane (where the cone base lies). For the root Z (t), however, we get

${\displaystyle M={\frac {1}{2}}\int \limits _{c}^{d}Z(t)\mathrm {d} t={\frac {1}{2}}\int \limits _{c}^{d}((p-u)\cdot {\dot {q}}-(q-v)\cdot {\dot {p}})\mathrm {d} t={\frac {1}{2}}\int \limits _{c}^{d}(p{\dot {q}}-{\dot {p}}q)\mathrm {d} t}$

because the definite integrals over the derivatives of uq and vp are zero. This follows from the constraints p (c) = p (d) and q (c) = q (d). From a geometrical point of view, this is the area of ​​the cone base . By partial integration (and observing p (c) q (c) = p (d) q (d)) one obtains the equation:

${\displaystyle M={\frac {1}{2}}\int \limits _{c}^{d}(p{\dot {q}}-{\dot {p}}q)\mathrm {d} t=\int \limits _{c}^{d}p{\dot {q}}\mathrm {d} t}$

The right expression impresses with its brevity, but it is impractical because the apparently complicated left expression is easier to evaluate. The area between the tangents from E to the cone base (not including the base itself), i.e. the area of ​​the tangent tip , results from

${\displaystyle M={\frac {1}{4}}\int \limits _{c}^{d}(|Z(t)|-Z(t))\mathrm {d} t}$

The factor ¼ (instead of ½) means that the area of ​​the tangent lobe is counted only once (instead of twice as with the folded cone surface, where the surface facing E and the surface facing away from E lie one above the other). If E lies on the edge or within the base of the cone, M disappears, namely then the base and the folded jacket fall into one.

From N

Ndt is the integration element of the perimeter of the cone base (see graphic). The scope of the cone base therefore results in

${\displaystyle U=\int \limits _{c}^{d}|N(t)|\mathrm {d} t}$

If one chooses only N (t) as integrand (instead of | N (t) |), it can happen that the integral vanishes. For example, the astroid (asteroid curve), the parameter representation p (t) = a cos (t) ³, q (t) = a sin (t) ³ over [0, ]. Then N (t) ² = 9a² sin (t) ² cos (t) ². For N (t) = 3a sin (t) cos (t) the integral vanishes over [0, ]. For | N (t) | however it results ${\displaystyle 2\Pi }$ ${\displaystyle 2\Pi }$

${\displaystyle U=3a\int \limits _{0}^{2\pi }\left|\sin(t)\cdot \cos(t)\right|\mathrm {d} t=12a\int \limits _{0}^{\frac {\pi }{2}}\sin(t)\cdot \cos(t)\mathrm {d} t=6a}$

From Z / N

The quotient measures the distance between the height base point E = (u, v) and the curve tangent at (p, q) as a function of t (see graphic). The general equation of the tangent to (p, q) is

${\displaystyle (p-x)\cdot {\dot {q}}-(q-y)\cdot {\dot {p}}=0}$

Division by N leads to the Hessian normal form. The distance between the point E = (u, v) and the tangent is obtained by inserting u and v in the normal form (without the zero): the result is Z / N. Example: The functions p (t) = r cos (t) + m and q (t) = r sin (t) + n over [0, ] describe the circle r around (m, n). Then Z (t) / N (t) = r + (mu) cos (t) + (nv) sin (t). When E moves into the center of the circle, i.e. when u = m and v = n, the result is Z (t) / N (t) = r, i.e. H. the perpendiculars from E to the circle tangents are the radius vectors of length r. ${\displaystyle 2\Pi }$

Example: slate circular cone

The parametric representation of the circle is: over . ${\displaystyle r}$ ${\displaystyle p(t)=r\cos(t),q(t)=r\sin(t)}$ ${\displaystyle [0,2\pi ]}$

If you put these values ​​and their derivatives in the formula for the mantle of the general oblique cone, you get the expression

${\displaystyle M={\frac {r}{2}}\int \limits _{0}^{2\pi }{\sqrt {(r-u\cdot \cos(t)-v\cdot \sin(t))^{2}+h^{2}}}\mathrm {d} t}$

With a suitable (fixed) angle , and can be represented as and , where , therefore, according to the addition theorem:, so that ${\displaystyle w}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle u=e\cos(w)}$ ${\displaystyle v=e\sin(w)}$ ${\displaystyle e^{2}:=u^{2}+v^{2}}$ ${\displaystyle u\cos(t)+v\sin(t)=e\cos(w-t)}$

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

When integrating over the full circle, the choice of does not matter. You can therefore bet. The integrand is for a function that is symmetrical in terms of function, so that one only needs to integrate over the semicircle and double the result, i.e.: ${\displaystyle w}$ ${\displaystyle w=0}$ ${\displaystyle w=0}$ ${\displaystyle \pi }$

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

See also

• Slate circular cone
• Oblique elliptical cone