CATENARY CURVE | USING MATLAB | 2D & 3D OUTPUT GRAPHS

CATENARY CURVE 

USING MATLAB

2D & 3D OUTPUT GRAPHS

Catenary is a famous curve in geometry and mathematics. Word caten is a latin origin which means of chain. Catenary is a U shaped curve and it is also referred with the name Alysoid and Chainette in some places. Visually it has a shape like a rope or cable which is hanging under its own weight due to earth’s gravitation i.e. saggy from the center when connected from only at its two ends.

Till year 1669 it was believed that the suspending rope connected from both ends which is handing upon its own weight holds the mathematical shape called parabola which is presented by Galileo. While in year 1669 his claim was disclaimed by Jungius while in year 1691 Leibniz, Huygens and Bernoulli presented the actual mathematical formula of catenary curve.

Mathematically the catenary curve is represented by hyperbolic cosine function.

y = a * cosh (x / a)

or :

y = (1/2) * a* ( e^(x/a)  +  e^(-x/a) )

where a is the factor which decides sagginess.

The catenary curve shape can be seen in suspension bridges, roof arches, overhead electric power cables and more.

In MATLAB we have coded the catenary curve both in 2D and in 3D formulations which provides two dimensional and three dimensional view of catenary curve shape. We also iterated the curve from higher to lower (i.e. increasing to decreasing sagginess) and showed each iteration using distinct color representation so that viewer gets the best visual understanding of the curve.

In code on the input side, we give rope left fixed end’s x and y coordinates; likewise for the right fixed end we have x and y coordinates.  Then user enters value of rope length as per his desired value. Also user has the leverage in the MATLAB code to construct a rope from N number of points decided by the user who runs the code.

The body of code constitutes of variables initialization, n-dimensions arrays and matrices formations, checking conditional statements, use of nested loops and implementation of mathematical formulations that are involved in obtaining the required outputs which are resulting values of x,y and z coordinates.

The output section of code after performing calculations of mathematical equations eventually gives us the resultant calculated values of x, y and z; which is graphically shown in the form of outputs plots in 2d multiple views and in 3d also with run time animation.