Advanced Math/Rectangle within a Circle Geometrical Shape.
Dear Prof Randy
Attached images are
1.Rectangle within a Circle.
2.Level Crossing - Intersection is rectangle shape highlighted by red line.
From Rectangle's center point draw a circle, here the rectangle edges will not touch the Circle boundary.
Is the Geometrical shape attached image 1 viz Rectangle within a Circle can be constructed geometrically ?.
i.e. is it a correct geometrical shape constructed similar to a square, rectangle, semicircle, circle, triangle, pentagon, parabola etc ?.
Awaiting your reply,
Thanks & Regards,
Prashant S Akerkar
The red "circle" drawn around the rectangle in your picture is not a circle but looks rather more like an ellipse. An honest-to-god ellipse is a "correct" geometrical shape in that it can be defined as the locus of points the sum of whose distance from 2 points is constant. That's a bit of a mouthful, but the easy, classic (fun) way to draw an ellipse is to put 2 thumbtacks in a piece of paper, separated horizontally by a couple of inches, and placed around them a loose loop of string. Then use a pencil to pull the loop taut and place its tip on the paper. Keeping the loop taut, draw a line around the tacks. The figure will be an ellipse.
There is a connection between squares/rectangles and circles/ellipses that you appear to want to exploit for esthetic reasons. A square has all sides equal whereas a rectangle can be defined as having just opposite sides equal (a square is a special case of a rectangle). A rectangle can thus be considered an elongated form of a square, with a long axis (say, horizontal axis, as in your example) and a shorter axis (vertical). Likewise, an ellipse is an elongated circle with different lengths between the horizontal and vertical edges (major and minor axes). Thus, having an ellipse surround a rectangle is esthetically consistent with a circle surrounding a square.
The mathematical definition of an ellipse is also iinteresting and revealing. It can be written, for x the horizontal axis and y the vertical, as
(x/a)^2 + (y/b)^2 = 1.
where a and b are constants and represent the lengths of the major and minor axes. The difference between the ellipse and the circle can also be seen in the fact that the circle has 1 "foci" or center, from which the points on the circle are all equidistant. The ellipse, on the other hand, has 2 foci, which, as it happens, correspond to the locations of the thumbtacks above. The distance between the foci (tacks) is given by 2c where
c^2 = a^2 + b^2.