Explaination PART B:

B.1):
Here the dimensionality of the system(after applying isomap) is 3. Here a significant drop is seen in residual variance for dimensionality 3, for k=7. Moreover , its quite clear from the motion of arm of nao from the images that there are only three degrees of freedom in its arm i.e. horizontal rotation at arm,elbow and wrist joints which constitutes 3 degrees of freedom.

B.2):The number of degrees of freedom for the given set of figures is four. On applying isomap on these images using eucleadian distance , we obtained continuously decreasing residual variance curve. However at dimensionality 4, the residual variance is very less (<0.5 10^(-4)) . Moreover here all the four decision vectors{theta1, theta2, theta3, theta4} are free to move in whole space giving 4 degrees of freedom, thus reducible dimensionality is 4.

B.3): Here:
theta 1 = 144.12 deg				theta 2=  48.22 deg

theta 3=  81.2 deg				theta 4=  290 deg

B.4):
Here in this case after applying onthe set of all images , we get the dimensionality of 1.
As in this case, the box moves along a horizontal line, restricting the decision vector i.e. {theta1, theta2, theta3, theta4} to unique value for every x-coordinate of center of box. The degree of freedom '1' determines that the decision vector is completely dependent on the position of box in horizontal line.
while in PartB.2) all the four vectors{theta1, theta2, theta3, theta4} are free to move in whole space.... thus their dimensionality is four.