A **cone** is a 3D surface generated by a moving straight line (called the **generator**) that always passes through a fixed point (the **vertex**) and intersects a given curve (the **guiding curve** or **base**).
This is a standard problem of finding the locus. We use the method of elimination.
Method:
V(α, β, γ).f(x, y) = 0 in the plane z = 0. (Example: x² + y² = a², z = 0).(x - α)/l = (y - β)/m = (z - γ)/n
z = 0.
z = 0 in the line equations:
(x - α)/l = (y - β)/m = (0 - γ)/n
x = α - lγ/ny = β - mγ/n(x, y, 0) must lie on the guiding curve f(x, y) = 0.
f(α - lγ/n, β - mγ/n) = 0
l, m, n using the original line equations (step 3).
l/n = (x - α)/(z - γ) and m/n = (y - β)/(z - γ)
f( α - (x - α)γ/(z - γ), β - (y - β)γ/(z - γ) ) = 0
α - (x - α)γ/(z - γ) = (αz - αγ - xγ + αγ)/(z - γ) = (αz - xγ)/(z - γ)β - (y - β)γ/(z - γ) = (βz - βγ - yγ + βγ)/(z - γ) = (βz - yγ)/(z - γ)The final equation of the cone is:
f( (αz - γx)/(z - γ), (βz - γy)/(z - γ) ) = 0
Question: Find the equation of the cone with vertex (α, β, γ) and base ax² + by² = 1, z = 0.
Solution:
Here f(x, y) = ax² + by² - 1 = 0.
Substituting the derived terms into f(..., ...) = 0:
a[ (αz - γx)/(z - γ) ]² + b[ (βz - γy)/(z - γ) ]² - 1 = 0
a(αz - γx)² + b(βz - γy)² = (z - γ)²A **right circular cone** is a special type of cone where:
All generators make a constant angle θ with the axis. This angle θ is called the **semi-vertical angle**.
The equation is derived from the constant angle property.
Let the cone have:
V(α, β, γ)(l, m, n) (Note: l² + m² + n² = 1)θLet P(x, y, z) be any point on the cone. The line segment VP is a generator.
The direction ratios of the generator VP are (x - α, y - β, z - γ).
The angle between the generator VP and the axis must be θ.
Using the angle formula cos(θ) = (a₁a₂ + b₁b₂ + c₁c₂) / (sqrt(a₁² + ...) * sqrt(a₂² + ...)):
cos(θ) = [ l(x - α) + m(y - β) + n(z - γ) ] / [ sqrt(l²+m²+n²) * sqrt((x-α)² + (y-β)² + (z-γ)²) ]
Since sqrt(l²+m²+n²) = 1, we can square both sides to get the final equation:
[ l(x - α) + m(y - β) + n(z - γ) ]² = cos²(θ) * [ (x - α)² + (y - β)² + (z - γ)² ]
(0,0,0) and the z-axis is the axis (l=0, m=0, n=1), the equation simplifies greatly:
[ 1(z) ]² = cos²(θ) * [ x² + y² + z² ]
z² = cos²(θ) * (x² + y² + z²)
z²(1 - cos²(θ)) = cos²(θ) * (x² + y²)
z² sin²(θ) = cos²(θ) * (x² + y²)
x² + y² = z² tan²(θ)
A **cylinder** is a 3D surface generated by a moving straight line (the **generator**) which remains parallel to a fixed line and intersects a given curve (the **guiding curve** or **base**).
The method is similar to the cone, but the generator equations are different.
Method:
x/l = y/m = z/n.f(x, y) = 0 in the plane z = 0.P(x₁, y₁, z₁) be *any* point on the cylinder.(x - x₁)/l = (y - y₁)/m = (z - z₁)/n
z = 0.
z = 0:
(x - x₁)/l = (y - y₁)/m = (0 - z₁)/n
x = x₁ - lz₁/ny = y₁ - mz₁/n(x, y, 0) must lie on the guiding curve f(x, y) = 0.
f(x₁ - lz₁/n, y₁ - mz₁/n) = 0
(x₁, y₁, z₁) to be on the cylinder. To get the locus, replace (x₁, y₁, z₁) with (x, y, z).
The final equation of the cylinder is:
f( x - lz/n, y - mz/n ) = 0
or
f( (nx - lz)/n, (ny - mz)/n ) = 0
Question: Find the equation of the cylinder whose generators are parallel to x = y = z and whose base is the ellipse x² + 2y² = 1, z = 0.
Solution:
1. The fixed line is x/1 = y/1 = z/1. So, l=1, m=1, n=1.
2. The base is f(x, y) = x² + 2y² - 1 = 0.
3. The locus is f(x - lz/n, y - mz/n) = 0.
4. Substitute values: f(x - 1z/1, y - 1z/1) = 0 => f(x - z, y - z) = 0.
5. Apply this to f:
(x - z)² + 2(y - z)² = 1A **right circular cylinder** is a cylinder where the guiding curve is a circle and the generators are perpendicular to the plane of the circle. The fixed line to which all generators are parallel is called the **axis** of the cylinder.
Alternatively, a right circular cylinder is the locus of a point in 3D space that is at a constant distance (the **radius**) from a fixed line (the **axis**).
This locus definition gives the easiest way to find the equation.
Let the cylinder have:
A(α, β, γ) with direction cosines (l, m, n).rLet P(x, y, z) be any point on the cylinder. The perpendicular distance from P to the axis must be equal to r.
We use a formula from 3D geometry: The perpendicular distance d from a point P to a line through A with d.c.s (l,m,n) is given by:
d² = (AP)² - (Projection of AP on the line)²
(AP)² = (x - α)² + (y - β)² + (z - γ)²l(x - α) + m(y - β) + n(z - γ)Setting d² = r², we get the equation of the cylinder:
r² = [ (x - α)² + (y - β)² + (z - γ)² ] - [ l(x - α) + m(y - β) + n(z - γ) ]²
(z - γ).(x - lz/n) and (y - mz/n).cos(θ) formula.