Smooth Parametric Curves and Their Slopes

Definition and Properties of Smoothness

A smooth parametric curve is one where the tangent line exists at every point and turns smoothly as the point moves along the curve. In contrast, a curve is not smooth at points where the tangent line does not exist, such as at cusps or corners.

• Smooth Curve: Tangent line exists everywhere; the curve transitions smoothly.

• Not Smooth Curve: At certain points (e.g., point P), the tangent line does not exist, indicating a cusp or corner.

Example: Cusp in Parametric Curves

Consider the parametric curve:

For : For : At , both derivatives vanish:

• as

• as

Although and , there is no tangent at due to the cusp at .

Theorem: Smoothness and Tangents of Parametric Curves

Conditions for Smoothness

Let be a parametric curve:

where and are continuous on .

• If on : The curve is smooth and has a tangent line at with slope .

• If on : The curve is smooth and has a normal line at with slope (the reciprocal of the tangent slope).

• If but : The curve has a vertical tangent.

Example: Horizontal and Vertical Tangents

For the curve:

Horizontal Tangent: and

Vertical Tangent: and

Example: Slope at a Specific Point

For the curve:

At :

Parametric Equations of Tangent and Normal Lines

General Formulas

Let be a smooth parametric curve at :

Tangent Line:

Normal Line:

Example: Tangent and Normal Lines at

For the curve:

At :

• Point:

• Derivatives: ,

• At : ,

Tangent Line:

Normal Line:

Concavity of Parametric Curves

Determining Concavity

The concavity of a parametric curve is determined by the sign of :

• : Curve is concave up

• : Curve is concave down

• : Curve has an inflection point

The formula for the second derivative is:

Example: Concavity and Tangents of a Parametric Curve

For the curve:

Key Points:

• :

• :

• Horizontal tangent:

• Vertical tangent:

• No cusps

• Concavity:

• Concave up:

• Concave down:

• No inflection points

• As :

• Symmetry: