An artist needs help from mathematicians! Angles of reflections: should I paint these distant trees in the water's reflections?

Solution 1:

In a simple mathematical model, the surface of the pond is a plane $P$; the viewer's eye is a point $E$ "above" $P$. A point $S$ in the scene is visible to $E$ as a reflection in the pond if the following conditions are met:

• There is a line segment from $S$ to a point $R$ (from which a light ray reflects) of the visible part of $P$ that does not hit any scene elements;
• The points $S$, $R$, and $E$ lie in a plane perpendicular to $P$, and the segments $SR$ and $RE$ make equal angles with $P$ at $R$:

The angle in the diagram is exaggerated for clarity, and it varies for different points in the scene. It may help to imagine yourself being at a point on the surface of the water looking up at angle $\theta$ toward the tall tree; do you see the top of the tree, or just the bank?

Solution 2:

Imagine that the surface of the water is part of a large flat mirror that extends through the earth and up to the horizon.

For each object you want to reflect, such as the large tree in the foreground, decide how high the base of it is above the level of the water, and imagine where the level of the mirror would be if it was underneath it.

Then the object is reflected in that mirror.