Difficulty: Intermediate | Prerequisites: Conic sections (circles, ellipses, parabolas, hyperbolas in 2D), basic 3D coordinate systems
Quadric surfaces are the 3D equivalents of conic sections. Just as circles, ellipses, parabolas and hyperbolas arise from slicing a cone in 2D, quadric surfaces arise from second-degree equations in x, y and z. They appear throughout Calculus III whenever you set up double or triple integrals, work with surface area, or describe physical objects mathematically. You should already be comfortable with conic sections and the 3D coordinate system.
There are six standard quadric surfaces (plus the sphere). Each is identified by its equation's structure and by the shapes of its cross-sections (called traces). Learn to match the equation form to the surface name, and you can sketch or identify any of them on an exam.
Quadric surface
Any surface in 3D defined by a second-degree polynomial in x, y and z. Think of it as the 3D version of a conic section.
Trace
The curve you get when you slice a surface with a plane (e.g. setting z = k). In simple terms, traces are the cross-sectional shapes that help you identify and sketch the surface.
Ellipsoid
A closed surface where every trace is an ellipse, defined by x²/a² + y²/b² + z²/c² = 1. Think of it as a stretched or squashed sphere.
Elliptic paraboloid
A bowl-shaped surface with elliptical horizontal traces and parabolic vertical traces. The variable raised to the first power (not squared) tells you which axis the bowl opens along.
Hyperbolic paraboloid
A saddle-shaped surface with hyperbolic traces in one direction and parabolic traces in another. Sometimes called a "saddle surface" or "Pringle shape."
Cone
A surface where horizontal traces are ellipses but vertical traces are hyperbolas (or pairs of lines at the vertex). Defined by z²/c² = x²/a² + y²/b².
Hyperboloid of one sheet
A surface that looks like a cooling tower, with elliptical horizontal traces and hyperbolic vertical traces. It has one connected piece and one negative coefficient in its equation.
Hyperboloid of two sheets
Two separate bowl-like pieces facing away from each other. It has two negative coefficients in its equation, and horizontal traces exist only beyond a certain distance from the origin.
Sphere
The set of all points at distance r from the centre. Equation: x² + y² + z² = r². A special case of the ellipsoid where a = b = c.
The fastest way to identify a surface is to count the squared terms and check their signs.
Three positive squared terms summing to 1: ellipsoid
Two positive squared terms equal to one first-power variable: elliptic paraboloid
Two squared terms with opposite signs equal to one first-power variable: hyperbolic paraboloid
Two positive squared terms equal to one negative squared term: cone
Three squared terms, two positive and one negative, equal to 1: hyperboloid of one sheet
Three squared terms, one positive and two negative, equal to 1: hyperboloid of two sheets
To find a trace, set one variable equal to a constant and see what curve remains.
Horizontal traces (set z = k): show the cross-section at height k
Vertical traces (set x = k or y = k): show the profile when slicing front-to-back or side-to-side
For example, the ellipsoid x²/a² + y²/b² + z²/c² = 1 gives an ellipse no matter which variable you fix. The cone gives ellipses horizontally but hyperbolas vertically (except at the vertex, where you get intersecting lines).
For paraboloids, the axis of symmetry is along the variable raised to the first power (not squared). For hyperboloids, the axis of symmetry is along the variable whose coefficient is negative.
The equation x² + y² + z² = r² is a sphere of radius r centred at the origin. This is an ellipsoid with a = b = c = r. If the equation has the form (x - h)² + (y - k)² + (z - l)² = r², the centre shifts to (h, k, l).
Surface | Standard Equation | Traces |
|---|---|---|
Sphere | x² + y² + z² = r² | All circles |
Ellipsoid | x²/a² + y²/b² + z²/c² = 1 | All ellipses |
Elliptic paraboloid | z/c = x²/a² + y²/b² | Horizontal: ellipses, Vertical: parabolas |
Hyperbolic paraboloid | z/c = x²/a² − y²/b² | Horizontal: hyperbolas, Vertical: parabolas |
Cone | z²/c² = x²/a² + y²/b² | Horizontal: ellipses, Vertical: hyperbolas (lines at vertex) |
Hyperboloid of one sheet | x²/a² + y²/b² − z²/c² = 1 | Horizontal: ellipses, Vertical: hyperbolas |
Hyperboloid of two sheets | −x²/a² − y²/b² + z²/c² = 1 | Horizontal: ellipses (for |
Students often confuse the hyperboloid of one sheet with the hyperboloid of two sheets. Count the minus signs: one negative coefficient means one sheet, two negative coefficients means two sheets.
Students sometimes think a cone has a "tip" at every point. The vertex (the point where the two halves meet) occurs only at the origin. Away from the vertex, the traces are smooth ellipses.
Students forget that the axis of symmetry for a paraboloid is determined by the variable that is NOT squared. If z is the first-power variable, the paraboloid opens along the z-axis.
Students mix up the elliptic paraboloid and the hyperbolic paraboloid. The elliptic version has both squared terms with the same sign (a bowl). The hyperbolic version has them with opposite signs (a saddle).
⚠️ Identifying a surface from its equation is a near-guaranteed exam question. Practise until the sign-counting is automatic.
⚠️ You will be asked to sketch traces. Set each variable to a constant in turn and name the resulting 2D curve.
⚠️ Completing the square is often needed to put an equation into standard form before you can identify it. If you see x² + 4x, rewrite as (x + 2)² − 4.
⚠️ Know that the hyperboloid of one sheet is the one that looks connected (cooling tower), and the hyperboloid of two sheets has a gap between its two pieces.
True or false: all traces of an ellipsoid are ellipses.
Fill in the blank: the axis of symmetry for a paraboloid is along the variable raised to the ______ power.
True or false: a hyperboloid of one sheet has two minus signs in its standard equation.
Fill in the blank: at the vertex of a cone, the vertical traces are ______ rather than hyperbolas.
True or false: x² + y² + z² = 9 describes a sphere of radius 9.
Q: Identify the surface given by x²/4 + y²/9 + z²/16 = 1.
A: Ellipsoid. Three positive squared terms summing to 1, with a = 2, b = 3, c = 4.
Q: What type of surface is z = x² + y²? Describe its traces.
A: Elliptic paraboloid (specifically circular, since the coefficients of x² and y² are equal). Horizontal traces (z = k, k > 0) are circles. Vertical traces are upward-opening parabolas.
Q: How do you distinguish between a hyperboloid of one sheet and a hyperboloid of two sheets from the equation alone?
A: Count the negative coefficients. One negative coefficient gives a hyperboloid of one sheet (one connected piece). Two negative coefficients give a hyperboloid of two sheets (two separate pieces).
Q: What surface is described by z² = x² + y²?
A: A cone. Horizontal traces are circles (for z ≠ 0). At z = 0, the trace degenerates to a single point (the vertex).
Q: Write the equation for a sphere centred at (1, −2, 3) with radius 5.
A: (x − 1)² + (y + 2)² + (z − 3)² = 25.
Quadric surfaces connect directly to double and triple integrals: when you set up bounds of integration in Calculus III, you are often integrating over or under one of these surfaces. They also appear in vector calculus when computing surface integrals and flux. The trace-sketching technique reappears whenever you need to visualise level surfaces of a function f(x, y, z) = c.
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