# Elliptic functions in parametric equations

• springo
In summary, the problem is to find the distance traveled by a point following a specific path, given by the equations x = 4·cn(λ), y = 3·sn(λ), z = am(λ), as λ increases. The elliptic functions involved are of the first kind with k2 = 1/2. The attempt at a solution involves finding the speed vector and setting it equal to (-4, 3, √2) to determine the value of λ at which the speed vector is parallel to the given vector. However, the next step is unclear and help is requested.

#### springo

Hey guys so I had trouble with this problem, I hope it's not too long and you can help me out.

## Homework Statement

A point moves following (starting at λ = 0 and increasing):
x = 4·cn(λ)
y = 3·sn(λ)
z= am(λ)
Find the distance the point has traveled when the speed vector is parallel to (-4, 3, √2).

## Homework Equations

The elliptic functions are of the first kind with k2 = 1/2.

## The Attempt at a Solution

I thought of the following:
v = dp/dt = (-4·sn(λ)·dn(λ), 3·cn(λ)·dn(λ), dn(λ))·(dλ/dt) = c·(-4, 3, √2) (with c a constant real number)
But then I don't know what to do next...

Thank you so much for your help!

Anyone could at least give me a clue on what my next move should be? Thanks.

## 1. What are elliptic functions in parametric equations?

Elliptic functions in parametric equations are mathematical functions that describe the motion of a point on an elliptic curve. They are defined by two parameters, which determine the shape and size of the curve, and can be used to represent various physical phenomena, such as planetary orbits and pendulum motion.

## 2. How are elliptic functions related to trigonometric functions?

Elliptic functions are closely related to trigonometric functions, as they are built from the inverse of the elliptic integral, which is a generalization of the inverse of the circular integral used in trigonometry. They share many of the same properties and can be used to solve similar problems.

## 3. What are some real-life applications of elliptic functions in parametric equations?

Elliptic functions in parametric equations have many practical applications, including in physics, engineering, and cryptography. They are used to model the motion of celestial bodies, design curves and surfaces in architecture, and create secure encryption algorithms.

## 4. What is the significance of the parameter in elliptic functions?

The parameter in elliptic functions represents the aspect ratio of the elliptic curve, which determines its shape and size. It is a key factor in understanding and manipulating these functions, and it can greatly influence the behavior of a system modeled by an elliptic function.

## 5. Are there any special properties or identities of elliptic functions in parametric equations?

Yes, there are many special properties and identities of elliptic functions that make them useful in various mathematical and scientific fields. For example, they have periodicity and symmetry properties, and they can be transformed into each other using modular transformations. They also have connections to other areas of mathematics, such as number theory and algebraic geometry.