Curves: lines, conics and parametric paths
A curve is the path of a moving point. A circle, an ellipse, a parabola and a hyperbola are the four conic sections: they appear when a flat plane slices a cone at different tilts. A level slice gives a circle, a slight tilt gives an ellipse, a slice parallel to the slant gives a parabola, and a steep slice that cuts both halves of a double cone gives a hyperbola. See Conic sections for their equations.
A curve can also be written by a parameter: x = f(t) and y = g(t). For example x = cos t, y = sin t draws a circle as t runs from 0 to 2π. This is the idea in Parametric equations.
Constructions: drawing exactly
A construction draws a shape using only a straightedge (unmarked ruler) and a compass. You can bisect an angle, draw a perpendicular bisector, copy an angle and build an equilateral triangle. Each step uses the fact that every point on a circle is the same distance from its centre. The details are in Geometric constructions.
Why it matters here: constructions are how the outline curves and cross-sections of surfaces were drawn long before computers.
Surfaces of revolution
A surface is a thin skin in 3D, with two directions along it. A surface of revolution is made by spinning a profile curve around a straight line called the axis. If the profile is at distance r(y) from the axis at height y, then every point of the surface satisfies x² + z² = r(y)².
Examples with the axis upright: a vertical line r = constant gives a cylinder; a slanted line r = k|y| gives a cone; a parabola gives a paraboloid (bowl); and a hyperbola gives a hyperboloid (cooling tower).
Slicing a surface (cross-sections)
Cut the surface with a flat plane at height y. For a surface of revolution, a cut across the axis is always a circle of radius r(y). Its area is πr². A cut along the axis shows the profile curve itself.
The same trick works for any solid: to understand a surface, slice it and look at the curves you get.
Key formulas and definitions
- Surface of revolution about the y-axis: x² + z² = r(y)²
- Cylinder: r(y) = c (constant)
- Cone: r(y) = k·y
- Circle slice: area = π·r(y)²
- Circle by parameter: x = cos t, y = sin t, 0 ≤ t ≤ 2π
- Ellipse by parameter: x = a cos t, y = b sin t
Worked examples
1. A vertical line 3 cm from the axis is spun round the axis. Name the surface and give the radius of a slice.
The profile is r = 3, a constant, so the surface is a cylinder. Every slice is a circle of radius 3 cm.
2. A cone has r(y) = 0.5y. Find the radius of the slice at y = 4.
r = 0.5 × 4 = 2. The slice is a circle of radius 2.
3. For the cone r = 0.5y, find the area of the slice at y = 4.
Area = π r² = π × 2² = 4π ≈ 12.57 square units.
4. A tower has r(y) = √(1 + y²) with y measured from the middle. Find the radius at the waist and at y = 3.
At y = 0, r = √1 = 1: the waist, the narrowest part. At y = 3, r = √(1 + 9) = √10 ≈ 3.16. It is wider at the ends.
5. A plane cuts a cone parallel to its slanted side. Which conic appears?
A parabola. A level cut gives a circle, a small tilt an ellipse, a cut parallel to the side a parabola, and a steep cut through both halves a hyperbola.
Common mistakes
- Thinking the profile curve and the surface are the same thing. The surface is what the curve sweeps when it spins.
- Forgetting that x² + z² = r² is a circle for each fixed y, not a single flat curve.
- Mixing up the cone and the paraboloid. A straight profile gives a cone; a curved parabola gives a bowl.
- Calling every sliced shape a circle. A cut across the axis is a circle, but a tilted cut of a cone can be an ellipse, parabola or hyperbola.