Various Dimensionality for the Manifold

We can observe that there is a sharp kink at dimensionality equal to 2.
So the Dimensionality of this manifold is best explained at the value 2.

2-D embedding when k=5 using Isomap.m

2-D embedding when k=5 using Isomap.m

2-D embedding when k=7 using Isomap.m

Θ1 varies along a circle around the toroid (in 2d), the values of Θ2 vary along a ring(crossection of the toroid) for a particular Θ1.

Θ2= 0 for outermost equitorial points, Θ2= +π and -Π for the innermost equatorial points.Θ2 varies along a straight line
for a particular theta1 on the 2D isomap, 0 at the outermost point, then +(Π/2) and -(Π/2) for points lying at the centre and +Π and -Π
for the innermost points.

3-D Space for k=5; View-1

View-2 for k=5

Path from image 0001.png to 0161.png in 2-D isomap embedding when the Obstacle 1 is present

Path from image 0001.png to 0161.png in 2-D Isomap embedding when Obstacle 2 is present superimposed with robot arms along the path.

The path in above figure appears to be similar to the path without an obstacle, no points in the shortest path
may collide with the obstacle in this case. Value of θ1 is not affected by the Obstacle region(green),
so the points in obstacle free region form the shortest path and avoid collision with the obstacle