Step 1: Set up the perspective-projection test.
A useful way to check whether a projection is "geometric" is to ask: can a light source be placed at some specific point relative to the globe and a screen (a plane, cone, or cylinder) tangent or secant to the globe, so that simply tracing straight light rays from the source through the globe's surface onto the screen reproduces this projection?
Step 2: Apply the test to the three azimuthal projections.
For the Gnomonic projection, put the light source at the centre of the sphere; the shadows of the graticule fall correctly onto a tangent plane, giving the Gnomonic graticule, this passes the test. For the Stereographic projection, put the light source at the point on the sphere opposite the tangent point; the resulting shadow pattern on the tangent plane is exactly the Stereographic graticule, this also passes. For the Orthographic projection, send the light rays in from infinity, parallel to each other and perpendicular to the tangent plane; the shadow pattern reproduces the Orthographic graticule, so this passes as well.
Step 3: Apply the test to Polyconic.
Trying to reproduce the Polyconic graticule this way fails. In the Polyconic projection every parallel is unrolled from its own separate tangent cone with a radius depending on that parallel's latitude, and the meridian spacing along each parallel is computed from the true great-circle distance along that parallel using trigonometric series, not from any single light source and single screen. Because a different cone (a different projection surface) is used for every parallel, there is no single geometric construction, no single light source and single developable surface, that generates the whole Polyconic graticule at once.
Step 4: Conclude the answer.
So among the four listed projections, only Polyconic fails the perspective-projection test, it is built up mathematically parallel-by-parallel rather than geometrically from rays and a single surface, making it the one that is NOT a geometric method of map projection.
\[ \boxed{\text{Polyconic}} \]
