Basic Parameters for Designing a "Macroscope"
Kouichi TOYOSHIMA
Faculty of Liberal Arts, University of Saga, Saga, 840, Japan
abstract
Several expressions for designing a "macroscope" arepresented. Limitations on positions of mirrors are also
discussed.

1. Introduction
In the preceding paper1), the author has introduced a simple optical
apparatus which enhances a 3-dimensional feeling for distant objects. Its structure
is nothing different from an ordinary mirror-type stereoscope designed to view alarge-sized pair of stereographs. What is new is its usage and scale. It is directed
to a real object, a landscape for example, to stare it by naked eyes with effectively
expanding their separation. I named this apparatus "macroscope".
In designing this apparatus it will be useful to provide some formulas toobtain the positions of mirrors and to optimize its main parameters, namely the
angular field of view and the effective baseline length.

1. Definitions
Definitions of positions are illustrated in fig.1. Since the apparatus has aleft-right symmetry, only the right half of its top view is shown. The center ofthe observer's eyes is taken to the origin of the Cartesian coordinate system. Theposition of the right eye is taken at (a,0). Objects are in the direction of they-axis. The positions of the inner and outer edges of the "eye" mirror are
denoted to P(x1, y1) and Q(x 1', y1'), respectively, and those of "objective" mirror
are R(x2, y2) and S(x2', y2'). For simplicity angle of these mirrors are fixed to 45
degrees to the x-axis. The position in the front-rear direction (y-axis) of the1"eye" mirror is indicated by the intersection of the extension of the "eye" mirror
plane with the y-axis,y0.
Mathematics used is only an elementary one but the calculation is somewhat
complicated and tedious.

ySeye mirror
QM1(0 , y0)

objective mirror

PP'

M2

xm

E(a , 0)x

R
R'
Fig.1. Right half of the apparatus is shown. E is the position of the right eye of theobserver, P(x 1 ,y 1 ) and Q(x 1 ',y 1 ') are the edges of the "eye" mirror, R(x 2 ,y 2 )and S(x 2 ',y 2 ') are those of "objective" mirrors. Angle of all mirrors are 45 degrees
to the x-axis. M1 and M2 are positions where the central ray (parallel to the y-axis) isreflected.

2. Effective baseline length and horizontal angular field of view
Throughout this note we consider the horizontal size of the apparatus be a
fixed parameter. The problem is to search an appropriate position (in the directionof y-axis) of two mirrors to give a wide angular field and largest possible
baseline length. In other words, for a given x2', the change in the baseline lengthand the horizontal visual angle are examined when y0 and y 2' is varied. Oncethese parameters are determined, positions P, Q and R are calculated using theleft-right symmetry of the visual angle.
2

Tangent of the right half of the visual angle α ism = x 1 '- ay 1'....(1).

Half of the baseline length (OX in fig.1) is proved to be l = a+y0-y2'+x2' after asimple calculation. The dependence of l on y0 and y2' is obvious, but to know thedependence of m on those parameters, we should calculate the position Q. Thisis readily known to be
Q ( - y 0 + (a +y 0 )(x 2 '+y 0 ) , (a +y 0 )(x 2 '+y 0 ) )a +x 2 '+2y 0 -y 2 ' a +x 2 '+2y 0 -y 2 '....(2)by using the fact that EQ and QS satisfies the reflecting condition with respect tothe "eye" mirror and Q(x1',y1') is on the line y = x +y0. Inserting this into (1) weobtainm = y 2 '-y 0 -ax 2 '+y 0....(3).

The dependence of l on y2' and y0 is that the larger the y2', the smaller thebaseline length, and contrary for y0. The parameter m is also nearly dependent
on y0-y2' because x 2' is much grater than ｜y0｜, but the sign is reversed.
Consequently these two parameters are in a competing relation; the larger thebaseline length, the smaller the angular field. So one should find a compromisedsolution.
In choosing the parameter y0, decreasing it reduces l but increases m . The
latter effect is much more significant and one should choose the smallest value
allowed by the geometrical interface to an observer's face.
Once a combination of y 0 and y2' is selected, the position P is determinedby the condition that the left half of the visual angle is the same as the right one,owing to the left-right symmetry of the apparatus. Using (2), one finds2

x1= a x 2 '+ 2a y 0 +y 0 -y 0 y 2 ' , y1= (a +y 0 )(x 2 '+y 0 )- a +x 2 '+y 2 '- a +x 2 '+y 2 'and for R,

3

22ax
2 '+ x 2 ' + 2ay0 + 2x 2 ' y 0 + 2y 0 - x 2 ' y 2 '- 2y 0 y 2 'x2 =,- a + x 2 '+ y 2 '22y2 = 2a x 2 '+ 2ay0 + 2x 2 ' y 0 + 2y 0 - ay2 '- x 2 ' y 2 '- 2y 0 y 2 '+y 2' .
- a + x 2 '+ y 2 '3. Limiting conditions to the horizontal visual field
In the above calculation, the following limitations should be considered iflarge values of y 2' or y0 are met.
(1) Limitation of left(or inner) half of the visual angle due to the left edge of theright eye mirror which cannot cross over the y-axis.
(2) The innermost ray which is to be reflected at R and P must not be interrupted
by the eye mirror.
The first condition means that the value a should not be smaller than they0right hand side of eq. (3). This immediately yields the following condition fory2' and y 0;2y 2 ' < ax2 '+2ay0 +y 0y0....(4).

If a small value of y0 is employed, this is automatically satisfied and this poses
practically no limitation. As an example, for a =3.2 cm and x2'= 80.25cm, theinequality (4) means the lower region of the curve in the y0-y2' plane, as shownin fig.24

250
200y 2' 150(cm)10050005101520y 0 (cm)Fig.2. Limitation of inner half of the visual angle by the inner edge of the eyemirror for a = 3.2cm and x2 '= 80.25cm. Below this curve is allowed.

To study the second condition, let us consider a ray which grazes theright edge(Q) of the eye mirror. This ray is reflected at R' and P', then arrive at
the right eye. The condition is that the angle(denoted to α ') of EP' to the y-axis
should be greater than or equal to that of EQ. Let m' = tan α' . Using (2) and thefact that EP' and P'R' as well as P'R' and R'Q satisfies the reflection condition,
m' is known to bem' =(a+x 2 '+2y 0-y 2 ') 2- (a+y 0 )(x 2 '+y 0 )(x 2 '+y 0-y 2 ')(a+x 2 '+2y 0-y 2 ')+(a+y 0 )(x 2 '+y 0 )....(5).

For the above combination of values of a and x2', the behavior of m' is plottedin fig. 3. The condition m =m' is shown by a curve in fig. 4 and below this curveis the allowed area(m >m'). As seen from these figures, the second condition issatisfied unless very large y2' is chosen.

570m'
0.960

40

y 2'

0.80.730(cm)50-2.5020

2.5y0 (cm)y2 '

540(cm)7.5
10-505

10

1015

20

2530

y0 (cm)Fig.3(left). m' is plotted against y0 and y2 ' for a = 3.2cm and x 2'= 80.25cm.
Fig.4(right). The region of m>m' is below the curve.

4. Design for given widths of the apparatus and the objective mirrors
It would be useful to provide formulas for optimizing the mirror positions
when the entire width of the apparatus and that of objective mirrors are given.
In this case the task is to find out an optimum value of y 2', and to calculate x1, y 1,x1' and y 1' for a given combination of x 2' and x 2.
In section 1, positions of Q, P and R are already obtained as a functionsof a, y0 and the coordinates of S. Then y0 is determined by use of the condition
that EP and EQ makes the same angle with EM1 , to bey0 =- a -x 2 '+y 2 ' ± - a+x 2 '+y 2 ' - a - 2 W+x 2 '+y 2 '2....(6)where W is the horizontal width of the objective mirror. Negative sign before
the square-root symbol should be discarded because it is an unphysical solutionthat the eye-mirror is put on the right to the eye, which means the staring line isdirected to the x-axis. Inserting this expression into eq (2) we can get theexpression for m;

m=- (a + W -x 2 ') t + (a - x 2 ')r2....(7).

(W -y 2 ') t - y 2 ' r
2The half of the baseline length l, on the other hand, is found to bel= s + r
2....(8).
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In these formulas the following replacements are employed to avoid lengthy
expressions:r = t (t- 2 W), s = a + x 2 '-y 2 ' , t = - a + x 2 '+y 2 ' .
( Expression (6) is rewritten as y0 = - s + r with these replacements.)
2An example of the behavior of y0, m and l are illustrated in fig.5 for a =3.2cm, x 2'= 80.25cm and W = 45cm. These graphs show that y 2' is mostly
determined to get a value of y0 around zero.

5. Vertical sizes of mirrors
Visual field angle in the vertical direction is limited by the sizes ofmirrors in that direction. Neccessary sizes of each edges, which give uniform
vertical field, are obtained by proportionating a height at each edge to distancesin x-coordinates from the image position (-y0, a+y 0) of the right eye by theeye-mirror. If we denote a height of the edge S to hS, heights at R, P and Q areas folows:
hR= x 2 +y 0 hS,x 2 '+y 0h Q= x 1 '+y 0 hS, hP = x 1 +y 0 hS .
x 2 '+y 0x 2 '+y 0

72010y0(cm)fig.5a0-1010203040

y 2' (cm)

0.35
610.3

m60l(cm)0.25

fig.5b

59
0.2
0

10

203040y 2 ' (cm)Fig.5. (a) Eye mirror position and (b) m, l are plotted against y2 ' for a = 3.2cm, x 2'=
80.25cm and W= 45cm.

I express my thanks to my sons Shinsaku and Kousaku who encouraged
this work by hastening me to make up this apparatus.
Reference1)

K. Toyoshima, Three-Dimensional Viewing of Distant Objects with"Naked" Eyes.

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