Assume a pole height h << R (radius of earth) erected at a point (long/lat $ =\theta,\phi).$
Depending on given time of day and month, is there a formula for shadow tip locus on ground (tangent plane)? We can make the most basic simplifying assumptions. Is the curve at 45$^0$ latitude more near to a conic ?
The line connecting the Sun with the Earth center describes a cone in a frame which is rotating with the Earth (we may neglect the relative motion of Earth and Sun during the short timespan of a day).
The same cone is described by the line connecting the Sun with the tip of your pole (because the Sun is very far away). The shadow of the tip is then located on the intersection between that cone and a plane (the ground), so it is by definition a conic section, usually a hyperbola.
That hyperbola changes slightly day after day, due to the relative motions of Earth and Sun.
EDIT
Set up a coordinate system on the ground, with the pole of length $h$ at the origin and the $x$-axis directed due North. The latitude $\theta$ is also the angle formed by the direction of the Earth rotation axis with the positive direction of the $x$-axis (I'm assuming we are in the North hemisphere, see the above diagram).
Suppose that at midday, when the Sun is due South, its direction forms an angle $\varphi$ with the negative direction of the $x$-axis. This angle of course changes day by day. Then the locus of the shadow tip is the hyperbola given by the equation $$ {(x-x_0)^2\over a^2}-{y^2\over b^2}=1, $$ where $a$, $b$ and $x_0$ can be found after some computation to be the following: $$ a={h\over2}\big(\cot\theta+\cot(2\varphi+\theta)\big),\quad b^2=a^2\cos^2\varphi \big(\tan^2(\varphi+\theta)-\tan^2\varphi\big), $$ $$ x_0={h\over2}\big(\cot\theta-\cot(2\varphi+\theta)\big). $$