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* 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,
Assumptions: the rod is parallel to x-axis
when flat.So it can vertically pulled in positive or negative
y-axis. |