P$13\mathrm{.}27$ The strain rosette shown in the figure was used to obtain normal strain data at a point on the free surface of a machine part.

$($a$)$ Determine the strain components ${}_x,{}_y,$ and ${}_{xy}$ at the point.

$($b$)$ Determine the principal strains and the maximum in$-$plane shear strain at the point.

$($c$)$ Draw a sketch showing the angle ${}_p,$ the principal strain deformations, and the maximum in$-$plane shear strain distortions.

$($d$)$ Determine the magnitude of the absolute maximum shear strain.

${}_a=510,{}_b=415,{}_c=430,v=0\mathrm{.}33$