Explicit, Implicit, Parametric, and Polar Curves

The Central Idea

A geometric curve is a collection of points arranged according to some relationship.

The same curve may be described by several kinds of equations:

• explicit: one coordinate is written directly in terms of another;
• implicit: the coordinates satisfy a relation;
• parametric: the coordinates are generated from a parameter;
• polar: each point is described by distance and direction from a pole.

The representation changes how the curve is described, but it does not necessarily change the geometric set of points.

A Curve as a Set of Points

In the Cartesian plane, a curve can be viewed as a set of ordered pairs

An equation or rule specifies which ordered pairs belong to the curve.

For example,

describes all points whose second coordinate equals the square of the first.

The resulting set of points forms a parabola.

Representation and Geometry Are Different Ideas

The geometry of a curve is the point set itself. A representation is the mathematical language used to describe that set.

For example, the unit circle can be written implicitly as

parametrically as

or in polar form as

Each representation emphasizes different structure.

Explicit Curves

An explicit Cartesian equation writes one coordinate directly in terms of another.

The most familiar form is

For each allowed input \(x\), the rule produces a corresponding value of \(y\).

Example: An Explicit Parabola

Consider

This is explicit because \(y\) has already been isolated.

Completing the square gives

so the vertex is

Explicit form makes evaluation easy because any chosen \(x\)-value can be substituted directly into the formula.

Explicit Does Not Mean Simple

An explicit equation may be algebraically complicated.

For example,

is still explicit because \(y\) is given directly as a function of \(x\).

The word explicit describes the form of the relationship, not its difficulty.

The Vertical-Line Restriction of \(y=f(x)\)

When a curve is represented as

each allowed \(x\)-value produces only one \(y\)-value.

Therefore, the graph must pass the vertical-line test.

Many important curves do not satisfy this requirement globally.

Why a Circle Is Not One Global Explicit Function \(y=f(x)\)

The unit circle satisfies

Solving for \(y\) gives

For most \(x\)-values between \(-1\) and \(1\), there are two possible \(y\)-values.

The circle can therefore be split into two explicit branches:

and

Explicit Form Can Use \(x=g(y)\)

Explicit form does not require \(y\) to be the dependent coordinate.

A relation may instead be written

For example,

describes a sideways parabola.

This curve fails the vertical-line test but passes the corresponding horizontal-line condition for \(x\) as a function of \(y\).

Implicit Curves

An implicit equation describes a relationship between the coordinates without necessarily isolating one of them.

A common general form is

Every point \((x,y)\) satisfying the relation belongs to the curve.

Example: The Unit Circle in Implicit Form

The equation

is implicit.

It treats \(x\) and \(y\) symmetrically and captures the entire circle in one equation.

This is one reason implicit equations are especially natural for closed curves.

Implicit Form Can Represent Multiple Branches at Once

Consider

Factoring gives

Therefore, the relation represents both lines

and

One implicit equation can therefore describe a curve with several branches or components.

Implicit Equations May Contain Singular or Special Points

An implicit equation can describe geometric behavior that is difficult to express with one ordinary function.

For example,

has a cusp at the origin.

Solving for \(y\) produces

for \(x\ge 0\), which splits the curve into two branches.

From Explicit to Implicit Form

An explicit equation can usually be rewritten implicitly by moving all terms to one side.

For example,

becomes

The two equations describe the same set of points.

From Implicit to Explicit Form

Sometimes an implicit relation can be solved for one coordinate.

Consider

Solving for \(y\) gives

The price of making \(y\) explicit is that one equation becomes two branches.

Not Every Implicit Relation Has a Convenient Explicit Form

Even when local explicit branches exist, solving an implicit equation algebraically may be inconvenient or impossible with elementary formulas.

For example,

is perfectly meaningful as an implicit relation even though isolating \(y\) is not necessarily useful.

Implicit form lets us study the geometry without requiring a closed-form solution for one coordinate.

Parametric Curves

A parametric curve describes both coordinates in terms of a third variable called a parameter.

In the plane, the standard form is

As \(t\) changes, the ordered pair

traces the curve.

The Parameter Is Not Necessarily Time

The letter \(t\) is often used because parameters frequently represent time in motion problems.

But a parameter can represent any convenient quantity.

We could equally write

or use any other parameter symbol.

Example: A Parametric Line

Consider

When \(t=0\),

When \(t=1\),

As \(t\) varies through all real numbers, the point moves along a line.

Eliminating the Parameter

A parametric curve can sometimes be converted into a Cartesian equation.

From

solve for \(t\):

Substitute into

Then

and therefore

Parametric Form Carries Orientation

A Cartesian equation describes which points lie on a curve.

A parametrization can additionally describe the order in which those points are traced.

For example,

traces the unit circle counterclockwise as \(t\) increases.

By contrast,

traces the same circle clockwise.

The Same Curve Can Have Many Parametrizations

The unit circle can be parametrized as

but also as

Both describe the same point set.

The second parametrization moves around the circle twice as quickly with respect to the parameter.

Parameter Intervals Matter

A parametrization is not fully specified by formulas alone when the allowed parameter values affect which part of the curve is traced.

For

the interval

traces the entire unit circle once.

The interval

traces only the upper semicircle.

Parametric Curves Can Describe Loops and Self-Intersections

Because \(x\) and \(y\) are generated independently from the parameter, parametric equations naturally describe shapes that may loop back or cross themselves.

For example,

produces a curve whose geometry is easier to understand parametrically than as a single explicit function.

Parametric Curves and Motion

Parametric equations connect naturally to motion and kinematics.

If

describes a moving point, then its position vector is

The curve is the path, while the parameter tells us how that path is traversed.

Polar Curves

A polar curve is described using polar coordinates

Here,

• \(r\) measures signed radial distance from the pole;
• \(\theta\) measures direction from the polar axis.

A common polar equation has the form

Polar Coordinates Describe Distance and Direction

In Cartesian coordinates, a point is located by horizontal and vertical displacement.

In polar coordinates, a point is located by radial distance and angle.

This makes polar form especially useful for geometry organized around a central point.

Cartesian and Polar Conversion Formulas

The coordinate systems are connected by

Squaring and adding gives

When \(x\neq 0\),

with the correct quadrant determined from the location of the point.

Example: A Circle in Polar Form

The equation

says that every point remains a distance \(3\) from the pole.

Therefore, it describes a circle of radius \(3\) centered at the origin.

Converting with

gives

Example: A Line in Polar Form

Consider

This fixes the direction while allowing radial distance to vary.

Since

we obtain

Thus the corresponding Cartesian line is

Negative Polar Radius

Polar coordinates allow negative values of \(r\).

A point with coordinates

lies in the direction opposite \(\theta\).

Equivalently,

represents the same point as

This is one reason polar coordinates are not unique.

Polar Coordinates Are Not Unique

The same point can have infinitely many polar coordinate descriptions.

For example,

and

represent the same point for any integer \(k\).

We also have

This differs from Cartesian coordinates, where an ordered pair identifies one point uniquely.

Polar Equations Naturally Describe Rotational Form

Curves with radial repetition, spiraling behavior, petals, or rotational symmetry often have compact polar equations.

Examples include

and

These families will appear again when we study generated curves and more specialized polar forms.

A Polar Equation Is Also Parametric

If a polar curve is written

then the Cartesian coordinates can be expressed as

Thus \(\theta\) acts as a parameter.

Every polar curve of the form \(r=f(\theta)\) therefore has a natural parametric representation.

Example: Converting a Polar Curve to Parametric Form

Suppose

Then

and

Therefore, one parametrization is

Example: Converting the Same Polar Curve to Cartesian Form

Start with

Multiply by \(r\):

Use

and

Then

Completing the square gives

The polar equation describes a circle of radius \(1\) centered at \((1,0)\).

One Curve, Four Viewpoints

Consider the upper semicircle of radius \(2\).

An explicit description is

An implicit description of the full containing circle is

together with the restriction

A parametric description is

A polar description is

Why Domain Restrictions Matter

Equivalent-looking equations may describe different point sets if their domains or parameter intervals are different.

For example,

gives

If \(t\) is allowed to range over all real numbers, the full sideways parabola is traced.

If \(t\ge 0\), only the half with

is traced.

Eliminating a Parameter Can Lose Traversal Information

Suppose

Eliminating \(t\) gives

The Cartesian equation tells us the geometric circle, but it no longer records:

• where the tracing begins;
• which direction the point moves;
• how rapidly the curve is traversed;
• how many times the curve is traced.

Converting Between Forms Is Not Always One-to-One

Different equations can describe the same geometric curve.

The unit circle can be written

or

The point set is the same, but the representations contain different information.

Choosing a Useful Representation

The best representation depends on what you want to see or calculate.

• Use explicit form when one variable naturally depends on another.
• Use implicit form when the coordinates participate symmetrically or several branches must be captured together.
• Use parametric form when order, motion, direction, or generated paths matter.
• Use polar form when radial distance, angle, rotation, or central symmetry organizes the geometry.

Explicit Form and Local Behavior

Even when a curve cannot be written globally as \(y=f(x)\), a portion of the curve may still admit an explicit description.

For the circle

the upper branch

is explicit for

This idea of working with local branches becomes increasingly important in advanced geometry and analysis.

Implicit Form and Level Sets

An equation such as

describes a level curve of the function \(F\).

For example,

is the level curve

for

Level curves and level surfaces will be developed later in this section.

Parametric Form and Vector-Valued Functions

A planar parametrization

can be packaged into a vector-valued function:

In three dimensions,

This connects curve geometry directly to the vector language developed earlier in Malone Math.

Curves in Three-Dimensional Space

Explicit and implicit descriptions become more subtle in three dimensions, but parametric form generalizes immediately.

A space curve can be written

For example,

describes a helix.

Its projection onto the \(xy\)-plane is the unit circle.

Implicit Equations in Three Dimensions Usually Describe Surfaces

In two variables, one equation

typically describes a curve.

In three variables, one equation

typically describes a surface.

For example,

describes the unit sphere.

This dimensional shift is central to the study of curves and surfaces.

Two Equations Can Define a Space Curve

In three dimensions, a curve can arise as the intersection of two surfaces.

For example,

and

together describe a curve lying on both a circular cylinder and a plane.

This is an implicit way to define a space curve.

Graphing an Explicit Curve

For

a practical graphing process is:

1. determine the domain;
2. identify intercepts;
3. look for symmetry;
4. locate important points;
5. analyze end behavior when relevant;
6. connect the information into a consistent curve.

Graphing an Implicit Curve

For

useful strategies include:

• solving for one variable when practical;
• checking symmetry under sign changes;
• finding intercepts;
• identifying familiar algebraic forms;
• examining restrictions on possible coordinates.

The goal is to understand the point relation, not necessarily to isolate one variable.

Graphing a Parametric Curve

For

it is useful to make a parameter table.

For example, with

selected values give

The resulting curve is the parabola

Graphing a Polar Curve

For

useful steps include:

1. choose important angles;
2. compute the corresponding radial distances;
3. plot the points using their directions;
4. watch for negative values of \(r\);
5. look for periodicity and symmetry;
6. connect the points according to the change in \(\theta\).

Symmetry Can Look Different in Different Representations

The circle

makes symmetry in \(x\) and \(y\) immediately visible.

The polar form

makes rotational symmetry even more direct because the radius is independent of angle.

Choosing a representation is often a way of choosing which symmetry becomes easiest to see.

Periodic Parameters Can Produce Closed Curves

If both coordinate functions repeat after some interval, the resulting parametric curve may close.

For example,

repeats every

This periodicity is exactly what allows the path to return to its starting point.

Closed Curves Are Not Necessarily Functions \(y=f(x)\)

A closed curve usually contains at least some vertical lines that intersect the curve more than once.

Therefore, many closed curves cannot be represented globally by one

Implicit, parametric, and polar forms are often better suited to them.

Self-Intersecting Curves

A self-intersection occurs when two different parts of a curve pass through the same point.

In parametric form, this can happen when different parameter values

produce the same point:

Parametric equations therefore provide a natural language for tracking how a path reaches the same location more than once.

Common Error: Assuming Every Curve Is \(y=f(x)\)

A curve is more general than the graph of a single real-valued function of \(x\).

Circles, sideways parabolas, loops, and many self-intersecting curves cannot be represented globally by one equation of the form

Use a representation appropriate to the geometry.

Common Error: Forgetting Branches

From

solving for \(y\) requires

Writing only

gives only the upper semicircle.

Common Error: Eliminating a Parameter Without Checking Restrictions

Suppose

Eliminating \(t\) gives

But the parameter restriction also gives

Without that restriction, the Cartesian equation would describe more of the curve than the parametrization actually traces.

Common Error: Treating the Parameter as a Coordinate

In

the parameter \(t\) usually helps generate the coordinates. It is not a third spatial coordinate unless the problem explicitly defines it that way.

The geometric point in the plane remains

Common Error: Assuming Polar Coordinates Are Unique

The points

and

are all the same point in the plane.

Polar equations must therefore be interpreted geometrically rather than as unique ordered-pair labels.

Common Error: Confusing \(r\) with \(x\)

In polar coordinates,

measures radial distance, not horizontal distance.

The horizontal coordinate is

The vertical coordinate is

Common Error: Assuming Equivalent Point Sets Carry Equivalent Information

The equations

and

describe the same geometric circle.

But the parametrization also provides an order of traversal.

Equivalent geometry does not imply identical representational information.

A Reliable Comparison Procedure

When comparing two curve representations:

1. identify the coordinate system being used;
2. determine the allowed domain or parameter interval;
3. convert forms when practical;
4. check whether every point in one representation appears in the other;
5. check whether extra branches or points were introduced;
6. note whether orientation or traversal information is preserved.

Example: Explicit to Parametric

Start with

A simple parametrization is obtained by letting

Then

Therefore,

Example: Implicit to Parametric

Start with the ellipse

Use the identity

A natural parametrization is

Substitution verifies the relation:

Example: Parametric to Implicit

Suppose

Then

and

Squaring and adding gives

Example: Cartesian to Polar

Convert

to polar form.

Use

and

Then

For the nonzero-radius part of the relation,

The pole \(r=0\) is already included by this polar equation for appropriate angles, so no geometric point is lost.

Why Dividing During Conversion Requires Care

In the previous example,

dividing by \(r\) assumes

Whenever a conversion divides by a variable or expression, check whether points corresponding to zero were excluded.

This is the same algebraic discipline used when solving equations and simplifying rational expressions.

Representations as Mathematical Tools

Different curve forms support different kinds of reasoning.

Explicit form emphasizes dependency.

Implicit form emphasizes constraint.

Parametric form emphasizes generation and traversal.

Polar form emphasizes radial and angular structure.

Learning to switch among these viewpoints is part of learning to see the same geometry from several mathematical directions.

Practice 1: Identify the Representation

Classify each equation as explicit, implicit, parametric, or polar.

(a)

(b)

(c)

(d)

Practice 2: Explicit to Implicit

Rewrite

in the form

Practice 3: Implicit to Explicit Branches

Solve

for \(y\).

State the domain of the resulting branches.

Practice 4: Eliminate a Parameter

Eliminate \(t\) from

Practice 5: Parametrize an Explicit Curve

Give a simple parametrization of

Practice 6: Parametric Circle

Eliminate the parameter from

Practice 7: Polar to Cartesian

Convert

to Cartesian form.

Practice 8: Cartesian to Polar

Convert

to polar form.

Practice 9: Restriction Matters

The parametrization

lies on the Cartesian curve

What part of the parabola is actually traced?

Practice 10: Same Curve, Different Traversal

Compare

with

Do they describe the same point set? What changes?

Practice Answers

Practice 1

(a) explicit,

(b) implicit,

(c) parametric,

(d) polar,

Practice 2

Move all terms to one side:

Practice 3

From

we obtain

and therefore

The domain is

Practice 4

From

solve for

Since

substitute:

Thus

Practice 5

Let

Then

Practice 6

Divide the coordinate equations by \(4\):

Squaring and adding gives

Therefore,

Practice 7

Since

the equation

becomes

Practice 8

Use

and

Then

The corresponding polar equation is

Practice 9

Because

and

we have

Therefore, only the right half of the parabola

is traced.

Practice 10

Both parametrizations trace the unit circle

The first traces the circle counterclockwise as \(t\) increases, while the second traces it clockwise.

What to Remember

A curve can often be described in several coordinate languages.

• Explicit: \[ y=f(x) \] or another coordinate is isolated directly.
• Implicit: \[ F(x,y)=0. \]
• Parametric: \[ x=x(t),\qquad y=y(t). \]
• Polar: \[ r=f(\theta). \]

The same geometric point set may have many representations, but the representations do not always preserve the same information.

The most important strategic question is: which representation makes the geometry easiest to understand?

Where We Go Next

These four representations give us a general language for describing curves. The next lesson, Conic Sections, studies one of the most important families of curves: circles, ellipses, parabolas, and hyperbolas.

We will compare their geometric definitions, equations, symmetries, focal properties, and coordinate representations.