2v

c

e

a

b

Průžez\ koulí

d

v

V_{válce}=\pi\cdot a^2\cdot 2v=2\pi v\(\left(\sqrt{r^2-v^2}   \right)} \)^2=\=2\pi v\left| r^2-v^2 \right| =2\pi v\left(r^2-v^2 \right)=\=2\pi vr^2-2\pi v^3

V_{vrchlíku}=\frac{1}{2}\pi ed^2+\frac{1}{6}\pi e^3=\frac{\pi}{2}\left(r-v\right)\(\left(\sqrt{r^2-v^2}\right)\)^2+\frac{\pi}{6}\(\left(r-v\right)\)^3=\=\frac{1}{2}\pi(r-v)(r^2-v^2)+\frac{\pi}{6}\left(r^2-2rv+v^2\right)(r-v)=\=\frac{\pi}{3}\left(2r^3-3r^2v+v^3)

Po\ odečtení\ objemů\ válce\ a\ dvou\ vrchlíků\ od\ zadané\ koule\ by\ měl\ (limitně)\ vycházet\ objem\ 
 koule\ s\ poloměrem\ [v].

V_{koule}-V_{válce}-2V_{vrchlíku}≟V_{koule\ s\ jiným\ poloměrem}\
\frac{4}{3}\pi r^3-\left(2\pi r^2v-2\pi v^3\right )-2\left(\frac{\pi}{3}\left( 2r^3-3r^2v+v^3 \right )   \right) ≟\frac{4}{3}\pi v^3

\frac{4}{3}\pi r^3-\left (\left(2\pi r^2v-2\pi v^3\right )+2\left(\frac{\pi}{3}\left( 2r^3-3r^2v+v^3 \right )   \right)\right)=\
=\frac{4}{3}\pi r^3- \left (2\pi r^2v-2\pi v^3+\frac{2}{3}\pi\left(2r^3-3r^2v+v^3\right)\right)=\
=\frac{4}{3}\pi r^3-\left(\frac{6\pi r^2v-6\pi v^3}{3}+\frac{2\pi\left(2r^3-3r^2v+v^3\right)}{3}\right)=\
=\frac{4}{3}\pi r^3-\left(\frac{6\pi r^2v-6\pi v^3+4\pi r^2v-6\pi r^2v+2\pi v^3}{3}\right)=\
=\frac{4}{3}\pi r^3-\left(\frac{4\pi r^3-4\pi v^3}{3}\right)=\frac{4}{3}\pi r^3-\left(\frac{4}{3}\pi r^3-\frac{4}{3}\pi v^3\right)=\
=\frac{4}{3}\pi v^3

a=\sqrt{r^{2}-v^2}

b=r-a=r-\sqrt{r^2-v^2}

c=v

d=\sqrt{r^2-c^2}=\sqrt{r^2-v^2}

e=r-c=r-v

Q.E.D.

Zadání:\ Dokažte,\ že\ objem\ tělesa\ vzniklého
\ vyvrtáním\ otvoru\ délky\ (2v)\ do\ koule\ má\ stejný\ objem\ bez\ ohledu\ na\ poloměr\ koule.\

(Osa\ vrtu\ prochází\ středem\ koule.)

V_{válce}

V_{vrchlíku}

S = (-1.0107781649245051, 0.017770034843202616, 0)
B = (1.8449073377541367, 0.08452675575319901, 0)
a = Sphere(S, B)
c = IntersectPath(xOyPlane, a)
f = Line(S, yAxis)
A, C = Intersect(f, a)
g = Segment(C, A)
h = Line(S, xAxis)
D, E = Intersect(h, a)
i = Segment(D, E)
F = Point(c)
j = Line(F, g)
G, H = Intersect(j, c)
k = Segment(F, H)
F' = Mirror(F, g)
H' = Mirror(H, g)
k' = Segment(F', H')
l = Segment(F, F')
m = Segment(H, H')
r = Segment(S, H')
I = Intersect(g, m)
vzdalenostSI = Distance(S, I)
b = Distance(S, I)
vzdalenostIC = Distance(I, C)
J = Intersect(i, k')
vzdalenostSJ = Distance(S, J)
vzdalenostJB = Distance(J, B)
vzdalenostIH' = Distance(I, H')
vzdalenostJH' = Distance(J, H')
d = OrthogonalPlane(I, g)
K = Intersect(g, l)
e = OrthogonalPlane(K, g)
p = IntersectPath(e, a)
q = IntersectPath(d, a)
L = Intersect(k', l)
n = Segment(K, L)
o, s, t, u = Cylinder(K, I, n)
v = true
w = true