Trig, Polar Coordinates & Holditch's Formula

With hindsight, Robinson's solution to the old problem of rigorous use of infinitesimal numbers in calculus boils down to Keisler's Funcrtion Extension Axiom that we discuss in Section 2.  Extending the (Dedekind) "real" numbers to include infinitesimals algebraically is not at all difficult, but calculus depends on small approximations with transcendental functions like sine, cosine, and the natural logarithm.  Following are some intuitive approximations with non-algebraic graphs.

Sine and cosine in radian measure give the and location of a point on the unit circle measured a distance along the circle as shown next.

Now make a small change in the angle and magnify by to make the change appear unit size.

Since magnification does not change lines, the radial segments from the origin to the tiny increment of the circle meet the circle at right angles and appear under magnification to be parallel lines.  Smoothness of the circle means that under powerful magnification, the circular increment appears to be a line.  The difference in values of the sine is the long vertical leg of the increment "triangle" above on the right.  The apparant hypotenuse with length is the circular increment.

Since the radial lines meet the circle at right angles the large triangle on the unit circle at the left with long leg and hypotenuse is similar to the increment triangle, giving

We write approximate similarity because the increment "triangle" actually has one circular side that is ≈-straight.  In any case, this is a convincing argument that .  A  similar geometric argument on the increment triangle shows that .

The Polar Area Differential

The derivation of sine and cosine is related to the area differential in polar coordinates.  If we take an angle increment of and magnify a view of the circular arc on a circle of radius , the length of the circular increment is , by similarity.

A magnified view of circles of radii and between the rays at angles and appears to be a rectangle with sides of lengths and .  If this were a true rectangle, its area would be , but it is only an approximate rectangle.  Technically, we can show that the area of this region is plus a term that is small compared with this infinitesimal,

Keisler's Infinite Sum Theorem assures us that we can neglect this size error and integrate with respect to .

Holditch's Formula

The area swept out by a tangent of length as it traverses an arbitrary convex curve in the plane is .  

We can see this interesting result by using a variation of polar coordinates and the infinitesimal polar area increment above.  Since the curve is convex, each tangent meets the curve at a unique angle, , and each point in the region swept out by the tangents is a distance along that tangent.

We look at an infinitesimal increment of the region in --coordinates, first holding the the -base point on the curve and changing .  Microscopically this produces an increment like the polar increment:

Next, looking at the base point of the tangent on the curve, moving to the correct -base point, moves along the curve.  Microscopically this looks like translation along the tangent line (by smoothness):

Including this near-translation in the infinitesimal area increment produces a parallelogram:

of height and base , or area :

Integrating gives the total area of the region

Created by Mathematica  (September 22, 2004)