(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

Emeli Hagan 2021-01-31 Answered

(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)24(x1)(y1)+5(y1)236=0 and (x2)2+6(y2)236=0. Find the transformation matrix between these two coordinate systems: (x1,y1) and (x2,y2).

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Expert Answer

l1koV
Answered 2021-02-01 Author has 100 answers

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]236=2X124X1Y1+5Y1236 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=12tan1(2HAB) 

Where H=COEFFICIENT OF X1Y1=4

A=COEFFICIENT OF X12=2 

B=COEFFICIENT OF Y12=5 

Solving we get P=0.89443Q=0.44721R=0.44721S=0.89443

Hence the transformation matrix is A=[0.894430.447210.447210.89443]

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