* Radius of curvature-r=360*l/2πθ  where θ is in degree or r=1/θ  where θ is in radian.

* Distance of extreme points from the center, d=(l/2)* sin(θ /2) / (θ /2). As θ --> 0, limiting value  of the expression in blue font becomes 1 and d=l/2;

    r*(θ/2) =l/2, sin (θ/2)=d/r; solving , we get the formula.

* Height of the extreme points above the initial flat surface, h= l*[Sin2(θ /4)] / θ  where θ is in radian. h=r - r*cos(θ /2) =2r * Sin2(θ /4) = 2(l/2θ)*Sin2(θ /4)

* Perpendiculars are drawn on the extreme points as shown in the figure. As the rod rotates, the perpendiculars also rotate. When the rod assumes the shape of a semi-circle, the perpendiculars make an angle 180 degree to each other and  parallel to the initial position of rod. They also rotate by an angle 90 degree. When the rod is fully bent to a circle, the perpendiculars make an angle 360 degree with each other, rotate by an angle 180 degree and are downward along the direction of vertical red line which is perpendicular at the center.

* The center of the circle of which the rod forms an arc, semi-circle or circumference of the full circle can be found out by extending the yellow perpendiculars till they meet at a point. It will be seen that this point lies along the red line, is at infinity when the rod is flat and descends down along the red line as the rod is bent and finally rests at a point when the full circle is made with mid point of the rod remaining stationary.

* it will be seen that the property of the curve has nothing to do with the surrounding surface.

* It seems  the flat surfaces are even dimensional,i.e.-0,2,4,6,....2n and curved surfaces are odd-dimensional-1,3,5,7,......(2n+1) even though curved surfaces can be fully specified by 2n co-ordinates. Curved surfaces are formed by projection of 2n dimensions into the (2n+1)th dimension.

* If B1 or A1 is curved in first quadrant, (x,y)=(+,+); second quadrant,(x,y)=(-,+); third quadrant,(x,y)=(-,-); fourth quadrant,(x,y)=(+,-) .

* If the co-ordinates of a point in a cartesian co-ordinate system is (x,y) and the axes are rotated by an angle θ or (-θ) , the new transformed co-ordinates (x',y') are

x'=xcosθ +ysinθ

y'=ycosθ -xsinθ

x=x'cosθ - y'sinθ 

y=x'sinθ +y'cosθ   

and sin(-θ) =-sinθ ; cos(-θ)=cosθ

In matrix form,

Length of line segment :(l)      
Angle (in degree) subtended at center :(thetad)      
Description Radius of Curvature Distance of extreme   

point from Center

Height of Extreme-point  from Initial Flat surface


chord length
Pure Flat Surface  
Curved Arc subtending an angle 90 degree at center (put angle=90)  
Curved arc (semi-circle) subtending angle 180 degree at center(put angle=180)  
Curved arc subtending angle 270 degree at center (put angle=270)  
Circle (put angle=360)  
General Formula with angle mentioned above