A line equation is given by: P = [3 3 0] + u[2 2 0] Find the end points of the l

Question

A line equation is given by: P = [3 3 0] + u[2 2 0] Find the end points of the line. Sketch the line with the end points and the u direction. Find the intersection points between the line and a circle with a center at (1, 2, 0) and radius of 2. Find the equation of the circle shown. Using this equation, find the coordinates of point P1 on the vertical centerline. Find the equation of the line that connects P1 and P2. What is the tangent vector of the line? Consider a Hermite curve in the xy plane defined by the following geometric coefficients: P(0)=(2, 3), P(1)=(4, 0), P'(0)=(3, 2), P'(1)=3, 4 Find a Bezier curve of degree 3 that represents the given Hermite curve as exactly as possible. In other words, determine the four control points of the Bezier curve. Expand both of the curve equations in polynomial form and compare them. The figure on the right shows two half circles C1 and C2 with centers at P1 and P2, respectively. Find the equation of the ruled surface that uses C1 and C2 as rails. Sketch the surface to show the u and v directions. Find the geometric center point of the rules surface using its equation.

Explanation / Answer

2.
The equation of the circle is given as :

(x-h)^2 + (y-k)^2 = r^2

here (h,k) is the center of the circle and r is the radius

in our problem we are given the center as (2,2) and the radius is r= 2

=> the equation of the circle is : (x-2)^2 + (y-2)^2 = 2^2

(x-2)^2 + (y-2)^2 = 4
from the given figure we could observe that
the point P1 is x = 2 and y =2+2 =4
=> P1= (2,4) and P2 = (8,4)
the equation of the line P1P2 is given as

(y-y1) = (y2-y1)/(x2-x1)*(x-x1)

y-4 = (4-4)/(8-4)*(x-2)
y - 4 = 0

=> the equation of the line is : y = 4

the tangent vector to the line y = 4 is :

v = y'/||y'|| = <0,0,0>/sqrt[0] = 0

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