Parametric Equations

Parametric equations are a way to describe curves and shapes using one or more parameters. Instead of expressing coordinates directly, we use these parameters to define how points move along the curve. This method offers flexibility in representing complex curves and analyzing their behaviour, making it useful in various fields like mathematics, physics, engineering, and computer graphics.

Parametric Equations Definition

A parametric equation is one where the x and y coordinates of the curve are both written as functions of another variable called a parameter; this is usually given the letter t or θ .

Parametric equations are sets of equations that show the position of a point using variables called parameters. These equations help describe how a point, curve, or surface moves or behaves in space. They offer more flexibility compared to regular equations, like those in graphs, because they allow us to express movement or changes more easily.

Types of Parametric Equations

There are many types of parametric equations, each describing different types of curves and shapes. Some common types of parametric equations are:

• Linear Parametric Equations: These equations represent straight lines in the form x(t) = at + b and y(t) = ct + d, where a and c are the slopes, and b and d are the intercepts.
• Circular Parametric Equations: These equations represent circles in the form x(t) = r⋅cos(t) and y(t) = r⋅sin(t), where r is the radius of the circle.
• Elliptical Parametric Equations: These equations represents ellipses in the form x(t) = a⋅cos(t) and y(t) = b⋅sin(t), where a and b are the lengths of the semi-major and semi-minor axes, respectively.
• Parametric Equations for Parametric Curves: These equations represent curves defined by arbitrary functions of a parameter t, such as x(t) = f(t) and y(t) = g(t), where f(t) and g(t) are any functions of t.
• Polar Parametric Equations: These equations represents curves in polar coordinates, often used for curves like cardioids, roses, and spirals, expressed as r(t) = f(θ) and θ(t) = g(t), where f(θ) and g(t) are functions of the angle θ.

Parametric Equations of Curves in Two Dimensions

Some of the common two dimensional curves with their parametric equation are given in the following table:

Curve	Normal Equation	Parametric Equation
Line	ax + by = c	x =x0​ +at and y = y0​ + bt
Circle	(x – h)2 + (y – k)2 = r2	x = r cos t + h and y = r sin t + k
Ellipse	(x – h)2/a2 + (y – k)2/b2 = 1	x = a cos t + h and y = b sin t + k
Parabola	Horizontal Parabola y – k = 4a(x – h)2 Vertical Parabola x – h = 4a(y – k)2	Horizontal Parabola x = at2 + h and y = bt + k Vertical Parabola x = at + k and y = bt2 + h
Hyperbola	(x – h)2/a2 – (y – k)2/b2 = 1	x = a sec t + h and y = b tan t + k
Cycloid	x = r arccos[(r-y)/r] – √(2ry – y2)	x = a(θ – sin θ) and y = a(1 – cos θ)
Lissajous Curve	–	x = a cos (k1t) and y = b sin (k2t)

Where,

• For the line, (x₀​, y₀​) is a point on the line, and a and b are the direction ratios.
• For the circle, (h, k) is the center of the circle and r is the radius.
• For the ellipse, (h, k) is the center of the ellipse, a is the length of the semi-major axis, b is the length of the semi-minor axis, and t is the parameter.
• For the parabola, (h, k) is the vertex of the parabola and a determines the direction of the opening.
• For the hyperbola, (h, k) is the center of the hyperbola, a is the distance from the center to a vertex along the x-axis, b is the distance from the center to a vertex along the y-axis, and t is the parameter.

Parametric Equations of Curves in Three Dimensions

Parametric equations of some of the three dimensional curves are given in the following table:

Curve	Parametric Equations
Line	x =x0​ +at, y = y0​ + bt, and z =z0​ +at
Plane	x = x0 ​+ at + bu, y = y0 ​+ ct + dv, and z = z0 ​+ et + fw
Sphere	x = h + rsin θ cos ϕ, y = k + rsin θ sin ϕ, and z = l + r cos θ
Ellipsoid	x = h + rcos θ sin ϕ, y = k + rsin θ sin ϕ, and z = l + r cos ϕ
Cylinder	x = h + rcos θ, y = k + rsin θ, and z = l + r cos θ
Cone	x = h + rcos θ, y = k + rsin θ, and z = l + r

Where,

• For the line, (x₀​, y₀​, z₀​) is a point on the line, and a, b, and c are the direction ratios.
• For the plane, (x₀​, y₀​, z₀​) is a point on the plane, and (a, b, c) and (u, v, w) are two direction vectors on the plane.
• For the sphere, (h, k, l) is the center of the sphere and r is the radius. θ is the polar angle and ϕ is the azimuthal angle.
• For the cylinder, (h, k, l​) is the center of the bottom circle, r is the radius, and c is the direction ratio along the z-axis.
• For the cone, (h, k, l​) is the tip of the cone, r is the radius of the base, H is the height, and θ is the angle between the side and the base.

Parametric Function Definition

A parametric function is a math rule where the output depends on one or more input variables, called parameters.

These functions help describe relationships between different quantities, like how something changes over time or with other factors.

Graphs of Parametric Function

The graphs of parametric functions are plots that show how the coordinates of points on a curve change as a parameter (often denoted as t) varies. These graphs typically display the relationship between the x and y coordinates of points on the curve over a specified range of the parameter.

The steps to create a graph of a parametric function are as follows:

Step 1 : Select a range for the parameter t. This range determines the curve you want to plot.

Step 2: Substitute different values of t into the parametric equations to calculate corresponding x and y coordinates.

Step 3: Plot each set of x and y coordinates on a coordinate plane.

Step 4: Connect the plotted points with a smooth curve to see the shape of the parametric curve.

For example, consider the parametric equations x(t) = 3cos(t) and y(t) = 3sin(t) making a circle of radius 1 centered at the origin.

By selecting a range for t (e.g., t from 0 to 2π), substitute various values of t into the equations to calculate corresponding x and y coordinates. Plotting these coordinates on a graph and connecting them with a curve will give the graph of the parametric function, which in this case will be a circle.

Applications of Parametric Equations

Parametric equations find applications in various fields. Some real-life applications where parametric equations are used are:

• Projectile Motion: Parametric equations are used to describe the motion of projectiles such as missiles, rockets, and thrown objects. For example, the trajectory of a thrown baseball can be modeled using parametric equations to predict its path.

• Robotic Arm Movement: Parametric equations are used to control the movement of robotic arms in manufacturing and assembly processes. For instance, the path of a robotic arm welding a car body can be described using parametric equations to ensure precise movements.

• Animation: Parametric equations are employed in computer graphics to create animations of moving objects. For example, the motion of a bouncing ball in an animated video game can be defined using parametric equations to control its position over time.

• Orbital Motion: Parametric equations are applied to describe the motion of celestial bodies such as planets, moons, and comets. For example, the orbit of a satellite around Earth can be modeled using parametric equations to calculate its position at different points in time.

• MRI and CT Scan Reconstruction: Parametric equations are used in medical imaging to reconstruct three-dimensional images from multiple two-dimensional scans. For instance, parametric equations can be employed to generate a 3D model of a patient’s internal organs based on MRI or CT scan data.

Read More,

• Equation in Maths
• Function
• Derivative of a Function in Parametric Form

FAQs on Parametric Equations

What is a curve described by the parametric equations?

A curve described by parametric equations is a path traced by a point whose coordinates are expressed as functions of one or more parameters. These equations define how the point moves over time or as the parameter changes.

What is the function of parametrization?

Parametrisation allows us to describe curves and shapes by specifying how each coordinate varies with respect to a parameter. This provides a flexible way to represent complex curves and study their properties.

How the curves are represented in parametric form?

Curves are represented in parametric form by defining equations for each coordinate (usually x and y) in terms of a parameter (often denoted as t or θ). These equations describe how the point moves along the curve as the parameter changes.

How do you write a parametric curve?

To write a parametric curve, you express the x and y coordinates as functions of a parameter t. For example, x(t)=cos(t) and y(t)=sin(t) represent a circle of radius 1 centered at the origin.

Why do we use parametric curves?

Parametric curves are used because they offer a flexible way to represent curves with complex shapes and behaviors. They allow us to study motion, analyze geometric properties, and solve problems in various fields such as mathematics, physics, engineering, and computer graphics.

What is the formula for parametric curve length?

The formula for the length of a parametric curve C from t=a to t=b is given by:

[Tex]L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt[/Tex]

where,

[Tex]\frac{dx}{dt} and \frac{dy}{dt}[/Tex] are the derivatives of x and y with respect to t, respectively. This formula calculates the distance traveled along the curve by integrating the rate of change of x and y with respect to t.