Geometric Construction 14
In this drawing:
show that the point Q is located at
.
Show that if
then h(b)=g(b),h(a)=f(a). If in addition we also have f(c)=g(c), then f(c)=h(c)=g(c).
(Aitken's Lemma)
Given Q1(x1,y1),Q2(x2,y2)
and Q3(x3,y3),
construct the graph of the polynomial p(x) satisfying
p(x1) = y1, p(x2)
= y2, p(x3) = y3.
Given (x1,y1), (x2,y2),
(x3,y3) and (x4,y4),
construct the graph of the polynomial p(x) satisfying
p(x1) = y1, p(x2)
= y2, p(x3) = y3,
p(x4) = y4 .
Given four points p0, p1,p2,p3,
we are to construct the Bezier cubic curve passing through these
points as follows:
-
Construct p01 = (1-t)p0 + tp1
-
Construct p12 = (1-t)p1 + tp2
-
Construct p23 = (1-t)p2 + tp3
-
Construct p012 = (1-t)p01 + tp12
-
Construct p123 = (1-t)p12 + tp23
-
Construct p0123 = (1-t)p012 + tp123
The locus of p0123 forms the required curve.
Show that
Construct the parabola tangent to two given straight lines at two given
points on each of the lines.
