Where Should You Stand to View the Mona Lisa?

An Exploration by Elizabeth Gieseking

The Marshall family has travelled to Paris to visit the Louvre and want to view the Mona Lisa. If Mary is 3 feet tall, Michael is 4 feet tall, Melissa is 5 feet tall and Max is 6 feet tall, how far away from the painting should each of them stand to get the largest viewing angle?

Viewing angle depends on the horizontal distance between the viewer and the painting, the vertical distance from eye level to the base of the painting, and the vertical distance from eye level to the base of the painting. We will start by considering the geometry of the situation.

In this figure, the painting is represented by the bold segement along the y-axis. The x-axis corresponds the the eye-level of the viewer. Point A is at the top of the painting, Point B is at the base of the painting. Points C and D correspond to positions of the viewer's eyes. Note that points C and D are both on the x-axis. This is due to the fact that the viewer's eyes are considered to be at a constant height for this problem. Let us consider angles ACB and ADB. The measure of an inscribed angle is half of the measure of the central angle which subtends the same arc on the circle. Since angles ACB and ADB subtend the same arc, their measures are the same. Let us now examine what happens when we change the viewing position.

In this image, if you move closer in from point D or further out from point C, the viewing angle decreases. Also note that the size of the circle increased. In both of these figures, the circle crosses the x-axis at two points. There is a point between our original points C and D in which the circle will only intersect the x-axis once - at the point of tangency. We will examine this graphically below.

These figures show the angles increasing and the size of the circle decreasing as C and D move closer together, reaching a maximum angle of 30.0 degrees when the circle is tangent to the x-axis and the points coincide.

Now we see that there is an optimum viewing position, but how can we determine it? We want to construct a circle through points A and B that is tangent to a line 2 feet below B. This circle will have its center along the line through the midpoint of AB and perpendicular to AB. We label this midpoint C. Because the distance |AB| = 4 and the perpendicular distance from C to the floor is also 4, we can construct a circle with center A and radius AB to find the center of the circle which will be tangent to the line y = 0, corresponding to eye level. We label this point D. Now we construct circle D with radius DA. This circle is tangent to y = 0 as desired.

We can calculate the optimum viewing angle from this diagram. We know |DA| = 4 from our construction and |AC| = 2. Therefore

The measure of an inscribed angle is half the measure of the central angle with the same endpoints. Therefore

The horizontal distance to the painting is given by

 

We will now turn to calculus for a derivation of the solution.

Let a = the vertical distance from eye level to the top of the painting

b = the vertical distance from eye level to the bottom of the painting

x = the horizontal distance from the viewer to the wall on which the painting is hung

= the angle from the viewer's eyes to the top of the painting

= the angle from the viewer's eyes to the bottom of the painting

Using the difference of angles formula for the tangent, we get:

These tangent values are the same as the slopes, so we can rewrite this equation as:

Now we can use calculus to find the maximum value of the tangent. The heights of the painting are constant, so we will just take the derivative with respect to x.

Since this is an optimization problem, we are looking for the point at which the derivative is zero. This occurs when

Thus the best viewing angle is obtained when standing at a distance from the painting. Note that this means the ideal distance will change with the viewer's height.

Now back to our question about viewing the Mona Lisa. The vertical dimension of the Mona Lisa is 30 inches and we will assume that the base of the painting is 6 feet or 72 inches from the floor. That means the top of the painting is 102 inches from the floor. We will also assume that each person's eye level is four inches less than their height.

Mary: 3 ft. tall

Eye level = 32 in.

a = 102 - 32 = 70 in.

b = 72 - 32 = 40 in.

Optimum distance =

Michael: 4 ft. tall

Eye level = 44 in.

a = 102 - 44 = 58 in.

b = 72 - 44 = 28 in.

Optimum distance =

Melissa: 5 ft. tall

Eye level = 56 in.

a = 102 - 56 = 46 in.

b = 72 - 56 = 16 in.

Optimum distance =

Max: 6 ft. tall

Eye level = 68 in.

a = 102 - 68 = 34 in.

b = 72 = 68 = 4 in.

Optimum distance =

So the next time you are in Paris, you can determine the optimum viewing distance for the Mona Lisa based on your height, that is, assuming there is no one standing in front of you.

 

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