A smooth parametrization of the semi circle which passes thr

Solution

The z coordinate of all 3 points is 5. Thus it can be assumed that the semi-circle is on the plane z =5;

Thus z =5.

Forgetting about the z part now,

it passes through (1,0) (0,1) and (-1,0). Plotting these 3 points , it can be seen they belong to a semicircle with centre at origin of radius 1.

Any circle of radius r can be parametrized as rcos(t) i + rsin(t) j

where as t varies from 0 to 2*pi the entire circle is traced out.

However for a semicircle we only need to vary t from 0 to pi, as we need only half the range of trace.

here r =1

Thus the parametrization is 1*cos(t) i + 1*sin/(t) j + 5 k

or cos(t) i + sin(t) j + 5 k (Ans)