(10%) In R2, there are two sets of coordinate systems, represented by two distinct bases: (x1,y1) and (x2,y2). If the equations of the same ellipse represented by the two distinct bases are described as follows, respectively: 2(x1)2−4(x1)(y1)+5(y1)2−36=0 and (x2)2+6(y2)2−36=0. Find the transformation matrix between these two coordinate systems: (x1,y1) and (x2,y2).

Question

l1koV · Accepted Answer

Let the transformation matrix be A=[PQRS]  A=[X1Y1]=[PQRS][X1Y1]=[PX1+QY1RX1+SY1]=[X2Y2] X2=[PX1+QY1] Y2=[RX1+SY1]  Putting in thhe given EQN. [PX1+QY1]2+6[RX1+SY1]2−36=2X12−4X1Y1+5Y12−36 P2+6R2=2 Q2+6S2=5 2PQ+12RS=−4......PQ+6RS=−2 Since the conversion is from a non homogeneus to homogeneous QN.,  Which is obtained by rotation through angle tyou can use the following transformation matrix as a formula [COS(T)SIN(T)−SIN(T)COS(T)][X1Y1]=[X2Y2]  Where T=12tan−1(2HA−B)   Where H=COEFFICIENT OF X1Y1=−4  A=COEFFICIENT OF X12=2   B=COEFFICIENT OF Y12=5   Solving we get P=0.89443……Q=0.44721……R=−−0.44721………S=0.89443  Hence the transformation matrix is A=[0.894430.44721−0.447210.89443]

(10%) In R^2, there are two sets of coordinate systems, represented by two distinct bases: (x_1, y_1) and (x_2, y_2). If the equations of the same ellipse represented

Answered question

Answer & Explanation

New Questions in Linear algebra