Parametric curves and conic paths: VCE Specialist Mathematics Units 3 & 4

1. A parameter describes how a point is selected

A Cartesian equation relates coordinates directly. Parametric equations describe each coordinate using another variable: x = f(t), y = g(t). Together they form the vector function r(t) = (f(t), g(t)). As t varies over its stated domain, the endpoint of the position vector traces a path. In three dimensions add z = h(t); the same parameter controls all coordinates.

The parameter may represent time, an angle or an abstract label. Do not assume time unless the context says so. A curve gives the set of positions, while a parametrisation also tells the order in which they are visited. When the parameter is time, it additionally encodes timing. Different vector functions can trace the same curve in different directions or at different speeds.

Sketch the endpoints of position vectors, rather than drawing every arrow from the origin. Useful sample values include endpoints of the parameter interval, exact trigonometric angles and values where a coordinate reverses direction. Label increasing-parameter arrows. Without them, a sketch of the Cartesian curve omits part of the information supplied by the parametrisation.

A parameter domain is not automatically the same as the x-domain. For example, a complete circle repeatedly visits most x-values. Nor does a coordinate range describe all possible pairs of coordinates: knowing separate x- and y-ranges is not enough to recover the curve. Coordinates must arise together from one permitted parameter value.

Prerequisite refresher: polar coordinates and loci

Polar coordinates describe a point by radial distance r and angle θ measured anticlockwise from the positive x-axis. Using the convention r ≥ 0, conversion gives x = r cos θ, y = r sin θ and r² = x² + y². A nonzero point has angles differing by whole turns; the origin has no unique angle. When recovering θ, use the coordinate signs to select the correct quadrant rather than relying on a tangent ratio alone.

For example, the circle x² + y² = 9 becomes r = 3 under this non-negative-radius convention. The line x = 2 becomes r cos θ = 2, or r = 2 sec θ. Choosing −π/2 < θ < π/2 gives positive r and traces the entire line once. The excluded angles correspond to directions parallel to the line, so division by cos θ there is invalid. Coordinate conversion needs domain checks just as parameter elimination does.

A locus is a set of points satisfying a geometric condition. A circle consists of points a fixed positive distance from its centre. A parabola consists of points equally distant from a fixed point, its focus, and a fixed line, its directrix; the focus is not on that line. An ellipse has a constant sum of distances from two fixed foci, with the constant greater than the distance between the foci. A non-degenerate hyperbola has a constant absolute difference of distances from two distinct foci, with that constant positive and smaller than their separation.

These definitions explain the geometric meaning behind familiar equations. The points equidistant from two distinct fixed points form their perpendicular-bisector line. Coordinate, polar and parametric descriptions express the same underlying locus, but a restriction may select only part of it. This refresher supplies the prerequisite language; the path analysis below keeps branch, orientation and timing explicit.

2. Eliminate the parameter without enlarging the path

When one coordinate can be solved conveniently for t, substitute that expression into the other coordinate. Then translate the parameter restrictions into coordinate restrictions. Elimination often produces a larger familiar curve, so the final answer needs the appropriate interval, branch or inequality. A correct equation with the wrong domain describes the wrong set of positions.

Squaring is especially risky. If x = t² and y = t with t ≥ 0, elimination gives x = y², but only y ≥ 0 is traced. The full sideways parabola would include negative y-values that the parameter forbids. Check a proposed Cartesian description in both directions: every parameter point must satisfy it, and every claimed curve point must come from an allowed parameter.

For circular functions, identities usually work better than solving an inverse function with multiple branches. From x − h = a cos t and y − k = b sin t, use cos²t + sin²t = 1. The resulting ellipse equation does not retain the parameter interval or orientation, so record those separately.

3. Circles, ellipses and hyperbolas from identities

A circle centred at (h, k) with radius R > 0 can be parametrised by (h + R cos t, k + R sin t). Taking 0 ≤ t < 2π visits every point once; including both endpoints of a closed interval repeats the starting point. Replacing t by −t reverses orientation. Replacing t by 2t doubles the number of circuits over the same parameter interval.

For an ellipse, use different nonzero scales on cosine and sine. Their absolute values give the semiaxis lengths. A negative scale reflects a coordinate and may reverse orientation, but does not make a length negative. If the scales have the same magnitude, the ellipse is a circle. Translating the coordinates changes the centre without changing the shape.

The identity sec²t − tan²t = 1 produces hyperbolas. With x = a sec t and y = b tan t, where a and b are positive, elimination gives x²/a² − y²/b² = 1. The parameter must exclude values where cos t = 0. Its allowed intervals determine which branch or branches are traced.

Not every hyperbola parametrisation uses circular functions. For instance x = h + t and y = k + c/t, with t ≠ 0 and c ≠ 0, gives (x − h)(y − k) = c. Positive and negative t select different branches. In either form, domain restrictions explain which asymptotic boundaries are approached and why those boundaries are not reached.

4. Three-dimensional curves require all three coordinates

A vector function r(t) = (2 cos t, 2 sin t, t) traces a helix. Its xy-projection is the circle x² + y² = 4, but z increases with t. After one complete turn, the x- and y-coordinates repeat while z rises by 2π. Therefore the spatial path is not a circle, and its points do not repeat after one turn.

Eliminating t to obtain x² + y² = 4 gives a cylinder containing the helix, not the helix itself. A single relation in three coordinates often describes a surface. Retain the vector or parametric description when it communicates the space curve more completely than attempted elimination.

When checking whether a point lies on a space curve, solve a convenient component for t and test every remaining component at that same value. A point can satisfy the cylinder equation and still fail the z-condition. Similarly, a projected crossing in a two-dimensional picture can correspond to two points at different heights.

5. Crossing paths and meeting particles are different conditions

For a geometric path intersection, solve r_A(s) = r_B(t) using separate parameters. This asks whether the paths share a position, possibly reached at different parameter values. For a collision with a shared time origin and time unit, solve r_A(t) = r_B(t), using the same time.

If one object starts later, translate its clock before comparing positions. For example, a formula given in elapsed time since its own departure must be evaluated at global time minus its departure time. Also enforce each object's interval of operation. A calculated meeting before launch or after motion ends is outside the model.

A useful non-routine check is to find path crossings first, then list each object's arrival time at every crossing. Equal positions alone are necessary but not sufficient for simultaneous meeting. The vector-calculus guide adds velocities and accelerations to these position constraints.

6. Use technology to inspect the parametrisation

Choose parametric graph mode and enter the exact parameter domain. Mark several known parameter points before trusting the graph. A default interval can accidentally add extra circuits, omit a branch or plot through a singularity. Plot separate permitted intervals on either side of an excluded value rather than joining them with a misleading segment.

A table of t, x(t), y(t) and, where relevant, z(t) helps check orientation. Sampling alone cannot prove that a branch is complete or that an endpoint is excluded; use identities, monotonicity and domain reasoning for those claims. Exact labelled points are more informative than an unlabelled screenshot.

To check a Cartesian result symbolically, substitute the parametric coordinates into its left side and simplify. Then separately check restrictions and coverage. An identity of one does not establish that the whole ellipse is visited. The algebra verifies containment, while the parameter analysis verifies precisely which part is attained.