Figure 30 Ist Scenario

x - is a distance between person and painting

h – is a height of the eye level of the person – 150 cm and 200 cm

L – as a height placement of the painting on the wall – 160 cm

l – is a difference between h and L – 10 cm or 40 cm more

Ph – is a size of the painting - 80

θ – viewer’s angle on the painting

β – viewer’s angle on the distance between painting and eye level

α – θ + β, angle from eye level to the highest point of the painting

Figure 30 represents a case when person’s height is 150 cm

Figure 31

By substituting values to the equation we will find a maximised distance for the person with 150 cm height

f (x) = Tan^-1 ( Tan^-1 (

f(x) = Tan^-1 ( Tan^-1 (

f(x) = Tan^-1 ( Tan^-1 (

Figure 31 clearly shows that Maximum point is between x=25 and x=35

f ‘(x) =

f ‘(x) =

Figure 32

when f ‘(x)=0

x= -30 and x=30 (figure 32)

We need only positive number as we need maximised value

So when distance from the picture is 30 cm person will see the picture with maximised angle, we will find this angle through already found x value.

θ = Tan^-1 ( Tan^-1 (

θ = 0.9272 Radians

θ = 0.9272 = 51.5662ﾟ

Figure 33

For the case when person is 200 cm high the angle and distance should be different because person’s height is bigger than a height placement of the painting on the wall. (figure 33)

In this case α is in front of the size of the painting – 80 cm

But other and very important point is that θ triangle has a straight angle but it is now not a full angle of the view

Figure 34

Height of the person is 200 cm, height placement is 160 cm, so person looks straight to the centre of the picture and this means that opposite side of the θ triangle is 40 cm

α = 2Tan^-1 (

F ormula for finding θ will be: (Figure 34)

θ = Tan^-1 (

Differentiated formula will be

θ’ =

β is a straight triangle as well and its opposite side is 40 cm

it is reasonable to state that α will be 2θ

α = 2 Tan^-1 (

α’=

α’ =

Figure 35

a t α’ =0 equation has an error

x	α’ =
-1	0.000083
0	undefined
1	0.000083

This means that the angle will decrease by moving from the painting and the optimised viewing angle is 180ﾟbut this is not possible in real life, so

At x=1 θ will be:

α = 2Tan^-1 (

α = 177ﾟ

At the case when person’s eye level height is 150 optimised distance from the painting is 30 cm and angle is 51.5ﾟ; for the case when person’s eye level is at some point at the painting maximised angle should be 180ﾟ, but it is not impossible so the least distance for the viewer can be 1 cm, at one cm the viewing and is 177ﾟ.

Figure 36

At the painting’s size 80 cm and 160 cm of height placement optimised distance for people with height from 150 cm to 200 cm is from 1 cm to 30 cm, the angle will be from 51.5ﾟ to 177ﾟ

IInd Scenario

x - is a distance between person and painting

h – is a height of the eye level of the person – 150 cm and 200 cm

L – as a height placement of the painting on the wall – 130 cm

Ph – is a size of the painting - 110

θ – viewers angle on the painting α + β

β – viewers angle on the distance between height placement of painting and eye level

α –angle from eye level to the highest point of the painting (figure 36)

An equation for the scenario two where eye level is on the painting will be:

θ= α + β

Opposite side for β will be h – L

For the case when person height is 150 opposite side for the β is:

150-130=20 cm

Figure 37

Opposite side for α will be the size of the painting – opposite side for β

1 10-20=90 cm

α= Tan^-1 (

β= Tan^-1 (

θ= Tan^-1 (

Figure 37 shows that the clearest viewing angle for the painting will be 180ﾟas in case 2 at the 1^st scenario.

Figure 38

For the person’s height 200 cm angle will have to change, but it might not happen due to the pattern when eye level is at the painting (figure 38)

Opposite side for β:

200-130=70 cm

Opposite side for α:

110-70=40 cm

α= Tan^-1 ( β= Tan^-1 (

x	θ ‘(x) =
-1	6502
0	undefined
1	6502

θ= Tan^-1 (

Table for the derived function provides the evidence that maximum angle for the viewing is 180ﾟ

Figure 39

To consider all cases and patterns I will investigate a situation when the height of the placement and eye level are the same:

x - is a distance between person and painting

h – is a height of the eye level of the person – 150 cm

L – as a height placement of the painting on the wall – 150 cm

Figure 40

P h – is a size of the painting - 60

θ – viewers angle on the painting

θ = Tan^-1 ( )

θ = Tan^-1 ( )

θ‘ =

The graph is the same as cases when eye level is on the painting, this means that optimised angle for this case is 180ﾟbut it is not possible in real life

At the case when person’s eye level height is 150 optimised distance from the painting is 30 cm and angle is 51.5ﾟ; for the case when person’s eye level is at some point at the painting maximised angle should be 180ﾟ, but it is not impossible so the least distance for the viewer can be 1 cm, at one cm the viewing and is 177ﾟ, for the cases wen person’s look goes straight at the painting optimised angle for it is 180ﾟ at zero distance.

At the cases when viewing person is lower them a height of the placement for 10 cm and when eye level is looking straight at a painting optimised distance for people with height from 150 cm to 200 cm is from 1 cm to 30 cm, the angle will be from 51.5ﾟ to 177ﾟ

Investigated data gives evidence that mathematical optimisation method is not reasonable for real-world situation due to impossible results as zero distance. For the museum keepers it is not advisable to let people approach the painting closer than 30 cm, but this distance is just the most optimised for a case when the painting is 80cm and it is 10 higher than eye level. The height of the placement should not be high due to standards, but 152 cm will not be proper for optimisation because average people’s height is 160 and optimised angle will be 180ﾟ which is not reasonable.