The Cone and the Cylinder

Cone with vertex at the origin

$ax^2 + b y^2 + c z^2 + 2 fyz + 2 gzx + 2hxy = 0 $

which is a homogenous equation.

Conversely, every homogenous equation of second degree in x, y, z represents a cone whose vertex is at the origin

Cor. : If the line $x/l = y/m = z/n$ is a generator of the cone (whose vertex is at the origin) $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$, then direction cosines (or direction ratios) l, m, n satisfy the equation of cone

$al^2 + b m^2 + c n^2 + 2fmn + 2gnl + 2hlm = 0$

Quadratic cone through the axes

The general equation of a cone of second degree passing through the three coordinate axes is


fyz + gzx + hxy = 0

Cone with a given vertex and given plane curve as the guiding curve

To find the equation of cone whose vertex is the point (α, β, γ) and base (guiding curve) the conic, $a x^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$



Required equation of cone:

$a (αz – γx)^2 + 2h (αz – γx) (βz – γy) + b (βz – γy)^2 + 2g (αz – γx) (z-y) + 2f (βz – γy) (z-γ) + c(z – γ)^2 = 0$

General second degree equation

Condition for the general equation of second degree to represent a cone and to find the coordinates of the vertex

$|\table a,h,g,u;h,b,f,v;g,f,c,w;u,v,w,d| $ = 0

Right circular cone

Equation of right circular cone

The equation of right circular cone whose vertex is at (α, β, γ), semi-vertical angle α and axis of the line ${x – α}/l = {y – β}/m = {z – γ}/n$

$[l (x – α) + m (y – β) + n(z – γ)]^2 = (l^2 + m^2 + n^2) [(x – α)^2 + (y – β)^2 + (z – γ)^2] cos^2 α$


which is the required equation of right circular cone

(A) Equation of right circular cone with vertex at origin

$(lx + my + nz)^2 = (l^2 + m^2 + n^2) (x^2 + y^2 + z^2) cos^2 α$

(B) Equation of right circular cone with vertex at origin and axis along z-axis

$x^2 + y^2 = z^2 tan^2 α $

Enveloping cone

Equation of the enveloping cone

To find the equation of the enveloping cone of the sphere $x^2 + y^2 + z^2 = a^2$ with vertex at the point (α, β, γ)



$[α (x – α) + β (y -β) + γ (z-γ)]^2 = [(x – α)^2 + (y-β)^2 + (z – γ)^2] (α^2 + β^2 + γ^2 – a^2)$

$S S_1 = T^2$

where,
$S_1 = α^2 + β^2 + γ^2 + 2uα + 2vβ + 2wγ + d$

$T = xα + yβ + zγ + u (x + α) + v(y + β) + w (z + γ) +d $

Right circular cylinder

Equation of a right circular cylinder

To find the equation of the right circular cylinder whose radius is r and axis is the line ${x-α}/l = {y-β}/m = {z-γ}/n$

$(x – α)^2 + (y -β)^2 + (z – γ)^2 – \{ {l (x – α) + m (y – β) + n (z – γ)}/{√{l^2 + m^2 + n^2}}\}^2 = r^2$

(A) Equation of a right circular cylinder whose axis is $x/l=y/m=z/n$

$(x^2 + y^2 + z^2) – \{ {lx + my + nz}/{√{l^2 + m^2 + n^2}} \}^2 = r^2$

(B) Equation of a right circular cylinder whose axis is z-axis

$x^2 + y^2 = r^2$

Enveloping cylinder

Equation of enveloping cylinder

To find the equation of the enveloping cylinder of the sphere $x^2 + y^2 + z^2 = a^2$ whose generators are parallel to the line $x/l = y/m = z/n$

$(lx + my + nz)^2 = (l^2 + m^2 + n^2) (x^2 + y^2 + z^2 – a^2)$