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PGD01C01
Module 4 · Vectors and Geometry of Space

Surfaces in Space; Cylindrical and Spherical Coordinates

Core Titles
Key headlines and terms for quick recall
  • Sphere x2+y2+z2=r2x^2 + y^2 + z^2 = r^2
  • Cylinder (e.g., x2+y2=r2x^2 + y^2 = r^2, zz free)
  • Cone, Quadric surfaces (ellipsoid, paraboloid, hyperboloid)
  • Cylindrical coordinates (r,θ,z)(r, \theta, z)
  • Spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi)
  • Conversions between Cartesian, cylindrical, spherical
Basic Idea
What it is, why it matters, how it works

Surfaces in space

Sphere of radius rr centred at (h,k,l)(h, k, l): (x−h)2+(y−k)2+(z−l)2=r2.(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2.

Cylinder. A surface generated by lines (rulings) parallel to a fixed axis. Standard right-circular cylinder along the zz-axis: x2+y2=r2x^2 + y^2 = r^2.

Cone. z2=x2+y2z^2 = x^2 + y^2 — a double cone with apex at origin.

Quadric surfaces — second-degree in three variables:

SurfaceEquation
Ellipsoidx2a2+y2b2+z2c2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1
Elliptic paraboloidz=x2a2+y2b2z = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}
Hyperboloid of one sheetx2a2+y2b2−z2c2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - \dfrac{z^2}{c^2} = 1
Hyperboloid of two sheets−x2a2−y2b2+z2c2=1-\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1
Hyperbolic paraboloid (saddle)z=x2a2−y2b2z = \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}

Cylindrical coordinates (r,θ,z)(r, \theta, z)

Replace (x,y)(x, y) with polar (r,θ)(r, \theta), keep zz. x=rcos⁡θ,y=rsin⁡θ,z=z.x = r \cos\theta, \quad y = r\sin\theta, \quad z = z. r≥0r \ge 0, θ∈[0,2π)\theta \in [0, 2\pi).

Inverse: r=x2+y2r = \sqrt{x^2 + y^2}, tan⁡θ=y/x\tan\theta = y/x.

Spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi)

ρ=\rho = distance from origin, ϕ=\phi = angle from positive zz-axis (polar angle, 0≤ϕ≤π0 \le \phi \le \pi), θ=\theta = azimuthal angle in xyxy-plane.

x=ρsin⁡ϕcos⁡θ,y=ρsin⁡ϕsin⁡θ,z=ρcos⁡ϕ.x = \rho \sin\phi \cos\theta, \quad y = \rho \sin\phi \sin\theta, \quad z = \rho \cos\phi.

Inverse: ρ=x2+y2+z2\rho = \sqrt{x^2 + y^2 + z^2}, cos⁡ϕ=z/ρ\cos\phi = z/\rho, tan⁡θ=y/x\tan\theta = y/x.

When to use which

  • Cylindrical — problems with axial symmetry (pipes, rotating bodies).
  • Spherical — problems with radial symmetry (planets, fields around a point source).

Why this matters in Data Science

Spherical coordinates appear in directional statistics, on-the-sphere embeddings, and geospatial data. Quadric surfaces describe error/loss landscapes for many models.

Mind Map
Visual structure of the concept
SURFACES & COORDINATES
├── Surfaces
│   ├── Sphere x² + y² + z² = r²
│   ├── Cylinder x² + y² = r²
│   ├── Cone z² = x² + y²
│   └── Quadrics: ellipsoid, paraboloid, hyperboloid, saddle
├── Cylindrical (r, θ, z)
│   ├── x = r cos θ,  y = r sin θ
│   └── Axial symmetry
└── Spherical (ρ, θ, φ)
    ├── x = ρ sin φ cos θ
    ├── y = ρ sin φ sin θ
    ├── z = ρ cos φ
    └── Radial symmetry
Exam Q&A
Part A (2 marks) and Part B (20 marks) style questions

Part A (2 marks each)

Q1. Write the equation of a sphere of radius 5 centred at (1,2,3)(1, 2, 3). (x−1)2+(y−2)2+(z−3)2=25(x-1)^2 + (y-2)^2 + (z-3)^2 = 25.

Q2. State the conversion from cylindrical to Cartesian. x=rcos⁡θ,  y=rsin⁡θ,  z=zx = r\cos\theta, \; y = r\sin\theta, \; z = z.

Q3. State the conversion from spherical to Cartesian. x=ρsin⁡ϕcos⁡θ,  y=ρsin⁡ϕsin⁡θ,  z=ρcos⁡ϕx = \rho \sin\phi \cos\theta, \; y = \rho \sin\phi \sin\theta, \; z = \rho \cos\phi.

Q4. Name two quadric surfaces. Ellipsoid and hyperboloid (also paraboloid, saddle).

Part B (20 marks)

Q. Describe the standard quadric surfaces. Define cylindrical and spherical coordinate systems and convert the point (1,1,2)(1, 1, \sqrt{2}) to both.

Quadric surfaces — examples and shapes.

SurfaceEquationShape
Spherex2+y2+z2=r2x^2+y^2+z^2=r^2round ball boundary
Ellipsoidx2a2+y2b2+z2c2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1stretched sphere
Elliptic paraboloidz=x2a2+y2b2z = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}bowl
Hyperboloid of one sheetx2a2+y2b2−z2c2=1\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1cooling-tower shape
Hyperboloid of two sheets−x2a2−y2b2+z2c2=1-\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1two cup-shaped sheets
Hyperbolic paraboloidz=x2a2−y2b2z = \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}saddle
Conez2=x2+y2z^2 = x^2 + y^2double cone
Cylinderx2+y2=r2x^2 + y^2 = r^2infinite tube

Cylindrical coordinates (r,θ,z)(r, \theta, z) — polar in the xyxy-plane plus zz: x=rcos⁡θ,y=rsin⁡θ,z=z,r=x2+y2,tan⁡θ=y/x.x = r\cos\theta, \quad y = r\sin\theta, \quad z = z, \quad r = \sqrt{x^2 + y^2}, \quad \tan\theta = y/x.

Spherical coordinates (ρ,θ,ϕ)(\rho, \theta, \phi): ρ\rho = distance from origin, ϕ\phi = polar angle from +z+z axis, θ\theta = azimuth: x=ρsin⁡ϕcos⁡θ,y=ρsin⁡ϕsin⁡θ,z=ρcos⁡ϕ.x = \rho \sin\phi \cos\theta, \quad y = \rho \sin\phi \sin\theta, \quad z = \rho \cos\phi. With ρ=x2+y2+z2,  cos⁡ϕ=z/ρ,  tan⁡θ=y/x\rho = \sqrt{x^2+y^2+z^2}, \; \cos\phi = z/\rho, \; \tan\theta = y/x.

Convert (1,1,2)(1, 1, \sqrt{2}).

Cylindrical: r=1+1=2r = \sqrt{1 + 1} = \sqrt{2}. tan⁡θ=1/1=1⇒θ=π/4\tan\theta = 1/1 = 1 \Rightarrow \theta = \pi/4. z=2z = \sqrt{2}. (r,θ,z)=(2,  π/4,  2).\boxed{(r, \theta, z) = (\sqrt{2}, \;\pi/4, \;\sqrt{2})}.

Spherical: ρ=1+1+2=2\rho = \sqrt{1 + 1 + 2} = 2. cos⁡ϕ=22⇒ϕ=π/4\cos\phi = \dfrac{\sqrt{2}}{2} \Rightarrow \phi = \pi/4. tan⁡θ=1⇒θ=π/4\tan\theta = 1 \Rightarrow \theta = \pi/4. (ρ,θ,ϕ)=(2,  π/4,  π/4).\boxed{(\rho, \theta, \phi) = (2, \;\pi/4, \;\pi/4)}.

Distances in SpaceVector-Valued Functions: Differentiation and Integration