Min and Max Irregular Tetrahedral Shadows

Problem #54

Tags: geometry puzzle

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This problem is intended to follow Min and Max Regular Tetrahedral Shadows.

Now let's imagine an opaque irregular tetrahedron, having edges of somewhat random lengths.
This tetrahedron may rotate along any axis, and as it does the shape and size of the shadow cast onto the ground changes.
Sometimes the changes can be quite dramatic, if the tetrahedron is highly irregular.

Problem Statement

Input data will contain the number of testcases T in the first line.
The following T lines will each contain 12 space-separated decimal values corresponding to xyz coordinates of the vertices of a tetrahedron, given in the format x1 y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4.

Answer
Should be a string of space-separated value pairs corresponding to the minimum and maximum possible areas of the shadow cast by the given tetrahedron.
Round all answers to the nearest integer.

Example

In this example each line with test-case is further sub-split into four lines (all except first indented) to simplify human reading.
To feed these data to your program you may need remove these extra line breaks (depending on your chosen language and input method).

input data:
124.505471 -169.257075 362.025425
    191.150081 -202.822632 395.169294
    117.056519 -223.908481 422.228445
    149.15761 -98.581877 484.601749
655.742148 -191.036593 -846.572974
    711.090649 -223.87769 -796.326856
    643.486942 -121.682616 -721.403079
    776.273121 -114.705966 -863.113431

answer:
2727 5863 3983 10283
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