@@ -290,37 +290,43 @@ def arc_area(self):
290290 &\hspace{1em}- \left( \sum_{j=0}^n P_j^{(1)} b_{j,n} \right)
291291 \left( n \sum_{k=0}^{n-1} (P_{k+1}^{(0)} -
292292 P_{k}^{(0)}) b_{j,n} \right)
293- dt
293+ dt,
294294
295- Where :math:`b_{\nu, n}(t) = {n \choose \nu} t^\nu {(1 - t)}^{n-\nu}`
295+ where :math:`b_{\nu, n}(t) = {n \choose \nu} t^\nu {(1 - t)}^{n-\nu}`
296296 is the :math:`\nu`'th Bernstein polynomial of degree :math:`n`.
297297
298298 Grouping :math:`t^l(1-t)^m` terms together for each :math:`l`,
299299 :math:`m`, we get that the integrand becomes
300300
301301 .. math::
302302
303- & \sum_{j=0}^n \sum_{k=0}^{n-1}
303+ \sum_{j=0}^n \sum_{k=0}^{n-1}
304304 {n \choose j} {{n - 1} \choose k}
305- [P_j^{(0)} (P_{k+1}^{(1)} - P_{k}^{(1)})
306- - P_j^{(1)} (P_{k+1}^{(0)} - P_{k}^{(0)})]
307- t^{j + k} {(1 - t)}^{2n - 1 - j - k}
308- \\
309- &= \sum_{j=0}^n \sum_{k=0}^{n-1}
305+ &\left[P_j^{(0)} (P_{k+1}^{(1)} - P_{k}^{(1)})
306+ - P_j^{(1)} (P_{k+1}^{(0)} - P_{k}^{(0)})\right] \\
307+ &\hspace{1em}\times{}t^{j + k} {(1 - t)}^{2n - 1 - j - k}
308+
309+ or just
310+
311+ .. math::
312+
313+ \sum_{j=0}^n \sum_{k=0}^{n-1}
310314 \frac{{n \choose j} {{n - 1} \choose k}}
311315 {{{2n - 1} \choose {j+k}}}
312316 [P_j^{(0)} (P_{k+1}^{(1)} - P_{k}^{(1)})
313317 - P_j^{(1)} (P_{k+1}^{(0)} - P_{k}^{(0)})]
314- b_{j+k,2n-1}(t)
318+ b_{j+k,2n-1}(t).
315319
316320 Interchanging sum and integral, and using the fact that :math:`\int_0^1
317321 b_{\nu, n}(t) dt = \frac{1}{n + 1}`, we conclude that the
318- original integral(:math:`\frac{1}{2}\int_0^1 B(t) \cdot n(t) dt`) can
322+ original integral can
319323 simply be written as
320324
321325 .. math::
322326
323- \frac{1}{4}\sum_{j=0}^n \sum_{k=0}^{n-1}
327+ \frac{1}{2}&\int_0^1 B(t) \cdot n(t) dt
328+ \\
329+ &= \frac{1}{4}\sum_{j=0}^n \sum_{k=0}^{n-1}
324330 \frac{{n \choose j} {{n - 1} \choose k}}
325331 {{{2n - 1} \choose {j+k}}}
326332 [P_j^{(0)} (P_{k+1}^{(1)} - P_{k}^{(1)})