This paper proposes a linear algorithm for metric reconstruction from projective reconstruction. Metric reconstruction problem is equivalent to estimating the projective transformation matrix that converts projective reconstruction to Euclidean reconstruction. We build a quadratic form from dual absolute conic projection equation with respect to the elements of the transformation matrix. The matrix of quadratic form of rank 2 is then eigen-decomposed to produce a linear estimate. The algorithm is applied to three different sets of real data and the results show a feasibility of the algorithm. Additionally, our comparison of results of the linear algorithm to results of bundle adjustment, applied to sets of synthetic image data having Gaussian image noise, shows reasonable error ranges.