Unit 5: Cones and Cylinders

1. Geometric Definition of a Cone

A cone is a surface generated by a moving straight line (called a generator) which passes through a fixed point (called the vertex) and continually intersects a fixed space curve (called the guiding curve).

2. Theorem: Cones with Vertex at the Origin

Theorem: Every homogeneous polynomial equation of degree two in $x, y, z$:

represents a cone whose vertex is at the origin $(0, 0, 0)$.

Proof:

Let $P(x_1, y_1, z_1)$ be any point lying on the surface, so that:

Consider the straight line connecting the origin $O(0, 0, 0)$ to $P(x_1, y_1, z_1)$. The coordinates of any arbitrary point $Q$ on this line are given by $(t x_1, t y_1, t z_1)$ for any scalar parameter $t \in \mathbb{R}$.

Substitute the coordinates of $Q$ into the left-hand side of the quadratic equation:

By equation (1), the bracketed expression is identically zero. Therefore:

This proves that every point on the line $OP$ lies entirely on the surface. Since this holds for every point $P$ on the surface, the surface is generated by a family of straight lines all passing through the origin. Hence, it is a cone with vertex $(0, 0, 0)$. $\blacksquare$

3. General Second-Degree Equation Representing a Cone

Consider the general quadratic equation:

If this represents a cone with vertex $V(\alpha, \beta, \gamma)$, shifting the origin to $(\alpha, \beta, \gamma)$ via $x = X + \alpha$, $y = Y + \beta$, $z = Z + \gamma$ must eliminate all linear and constant terms, yielding a homogeneous quadratic in $X, Y, Z$.

This requirement is equivalent to the simultaneous vanishing of all partial derivatives at the vertex:

For these four linear equations in $(\alpha, \beta, \gamma, 1)$ to have a consistent non-trivial solution, the $4 \times 4$ determinant of coefficients must vanish:

This is the necessary and sufficient condition for the general quadratic equation to represent a cone.

1. Geometric Definition

A right circular cone is a surface generated by a line passing through a fixed vertex, maintaining a constant angle $\alpha$ (the semi-vertical angle) with a fixed line through the vertex (the axis).

2. Analytical Derivation of the Equation

Let:

• Vertex: $V(\alpha_v, \beta_v, \gamma_v)$
• Axis: Line passing through $V$ with direction cosines $(l, m, n)$ (where $l^2 + m^2 + n^2 = 1$)
• Semi-vertical angle: $\alpha$ ($0 < \alpha < \frac{\pi}{2}$)

Let $P(x, y, z)$ be any point on the cone. The generator line $VP$ has direction ratios:

The cosine of the angle between $\vec{VP}$ and the axis vector $\vec{A} = (l, m, n)$ is $\cos\alpha$:

Squaring both sides and setting $l^2 + m^2 + n^2 = 1$:

Canonical Case (Vertex at Origin, Axis along Z-Axis):

If $V = (0, 0, 0)$ and the axis is the $z$-axis $(l=0, m=0, n=1)$:

This is the familiar standard equation of a right circular cone.

1. The General Cylinder

A cylinder is a surface generated by a straight line (the generator) that moves parallel to a fixed given direction and continually intersects a fixed guiding curve.

Equation with Generators Parallel to Given Line:

Let the generators have direction ratios $(l, m, n)$, and let the guiding curve be a plane conic in $z = 0$:

Let $P(x, y, z)$ be any point on the cylinder. The generator through $P$ has equations:

This line pierces the plane $Z = 0$ when:

Since $(X, Y, 0)$ lies on the guiding curve $f(X, Y) = 0$, substitute $X$ and $Y$:

Expanding this gives the Cartesian equation of the cylinder.

2. Right Circular Cylinder

A cylinder whose guiding curve is a circle and whose generators are perpendicular to the plane of the circle is a right circular cylinder.

If the axis is the line $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$ (with $l^2+m^2+n^2=1$) and radius is $r$, the perpendicular distance from any point $P(x, y, z)$ to the axis equals $r$:

3. Enveloping Cone and Enveloping Cylinder of a Sphere

Enveloping Cone:

The locus of tangent lines drawn from an external point $V(x_1, y_1, z_1)$ to a sphere $S: x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$ is called the enveloping cone.

By Joachimsthal's quadratic method, if a line through $V$ touches the sphere, the quadratic equation for the intersection points has equal roots (discriminant $\Delta = 0$). This yields the fundamental relation:

where:

• $S = x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d$
• $S_1 = x_1^2 + y_1^2 + z_1^2 + 2ux_1 + 2vy_1 + 2wz_1 + d$
• $T = x x_1 + y y_1 + z z_1 + u(x + x_1) + v(y + y_1) + w(z + z_1) + d$

Enveloping Cylinder:

Similarly, the locus of tangent lines to a sphere $S = 0$ parallel to a fixed direction $(l, m, n)$ is the enveloping cylinder of the sphere.