In mathematics, a curve is generally defined as a continuous and connected set of points. It can be thought of as the path traced by a moving point.

Curves can be simple (not crossing themselves), closed (start and end at the same point), or both.

While a straight line is a type of curve, the term 'curve' often implies a path that is not straight.

Curves can be classified in various ways:

• Planar Curves: Curves that lie entirely within a single plane (e.g., circles, ellipses, parabolas, hyperbolas).
• Space Curves: Curves that exist in three-dimensional space and cannot be confined to a single plane (e.g., helix, knot curves).
• Algebraic Curves: Defined by polynomial equations (e.g., conic sections).
• Transcendental Curves: Defined by non-polynomial equations involving trigonometric, exponential, or logarithmic functions (e.g., cycloid, catenary).
• Open Curves: Have distinct start and end points (e.g., parabola, line segment).
• Closed Curves: Start and end at the same point, forming a loop (e.g., circle, ellipse).
• Simple Curves: Do not intersect themselves.
• Non-Simple Curves: Intersect themselves (e.g., figure-eight curve).

Curves can be represented using different mathematical forms:

• Explicit Form: \(y = f(x)\) (for 2D curves), or \(z = f(x, y)\) (for 3D surfaces that are functions of two variables).
• Implicit Form: \(F(x, y) = 0\) (for 2D curves), or \(F(x, y, z) = 0\) (for 3D surfaces). For example, a circle \(x^2 + y^2 - r^2 = 0\).
• Parametric Form: Coordinates are expressed as functions of one or more parameters. For a 2D curve: \(x = f(t)\), \(y = g(t)\). For a 3D curve: \(x = f(t)\), \(y = g(t)\), \(z = h(t)\). This is very powerful for describing complex paths and motions.

Key properties used to analyze curves include:

• Length (Arc Length): The total distance along the curve. Calculated using integrals.
• Curvature: How sharply a curve bends at a given point.
• Tangent Line: A line that touches the curve at a single point and has the same slope as the curve at that point.
• Normal Line: A line perpendicular to the tangent line at a point on the curve.

Curves are indispensable in:

• Physics: Describing trajectories, wave forms, and orbits.
• Engineering: Designing roads, bridges, airplane wings, and fluid flow systems.
• Computer Graphics and CAD: Modeling smooth surfaces, fonts, and animations (e.g., Bézier curves, splines).
• Art and Design: Creating aesthetically pleasing shapes and patterns.
• Biology: Analyzing natural shapes like DNA helices and leaf veins.