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Description  |
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BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention is in the field of non-perverting single surface reflecting
mirrors.
2. Description of the Prior Art
Most mirrors used today such as plane, convex, concave and parabolic
mirrors give perverted images i.e., the image of a right hand is a left
hand, images of words are backwards, and so forth. It is possible to
obtain non-perverted images by utilizing two plane mirrors at right angles
to effect a double reflection. Two successive image-perversions combine to
produce a non-perverted image. An example of such a technique is shown in
U.S. Pat. No. 1,991,054 to Hampke.
A non-perverted image may also be formed with a concave cylindrical mirror.
Such a cylindrical mirror consists of a single reflecting surface and is
referred to in classic literature of Plato and Lucretius, and is described
in modern works such as Mathematical Puzzles and Diversions, by Martin
Gardner, Simon and Schuster, New York (1959).
It is desirable to provide a non-perverted single surface mirror which
gives a true image of the object under observation. The disadvantage of
the prior art non-perverted concave cylindrical surface is the large image
distortion present resulting from elongation of the image in the direction
of the cylindrical axis. An additional problem with the concave
cylindrical mirror is its poor image focusing ability inasmuch as
cylindrical rather than spherical wavefronts are generated.
SUMMARY OF THE INVENTION
It is an object of the invention to provide a non-perverting mirror which
is free from the disadvantages of the prior art and which comprises a
single, continuous reflecting surface.
Yet another object of the invention is to provide a single surface
non-perverting mirror which provides a substantially non-distorted image
of the object under observation.
A further object of the invention is to provide a single surface
non-perverting mirror which reflects substantially spherical reflected
wavefronts to provide an image which may be sharply focused.
Another object of the invention is to provide a non-perverting mirror
having a number of saddle points to provide non-perverted images having
advantageous rotation properties upon rotation of the mirror and/or
object.
The mirror of the instant invention thus provides non-perverted images of
objects and comprises a light reflecting surface having at least one
intersecting concave and convex region. The intersecting regions define a
surface region of the mirror which has spacially concident concave and
convex curvatures. In the preferred embodiment of the invention the
direction of concave curvature is substantially orthogonal to the
direction of convex curvature.
The mirror may be used as a novelty device or in optical instruments where
a non-perverting image is desired. Means for rotating the non-perverting
mirrors may also be provided to take advantage of the image rotation
properties of the mirror so that large image rotations are achieved with
only a small corresponding mirror rotation.
BRIEF DESCRIPTION OF THE DRAWINGS
These and other objects and advantages of the invention will become clear
from the following detailed description taken in conjunction with the
drawings wherein:
FIG. 1A is a perspective view of a prior art concave cylindrical reflecting
surface;
FIG. 1B is a diagramatic representation of a non-perverting image such as
produced by the reflecting surface of FIG. 1A;
FIGS. 2A-2C are diagramatic representations of reflecting wavefronts from
the concave cylindrical surface of FIG. 1;
FIG. 3 is a partial view of a torus showing a segment thereof forming the
toroidal surface of the invention;
FIG. 4 is an enlarged view of a torodial mirror in accordance with the
invention;
FIGS. 5A-5D are diagramatic representations of reflecting wavefronts from
the toroidal mirror of the invention;
FIGS. 6A-6C are diagramatic representations of ray diagrams for convex,
planar and concave surfaces respectively;
FIG. 7 shows a two nodal, saddle shaped surface in accordance with the
invention;
FIGS. 8A-8C are perspective drawings of one embodiment of the invention
showing a method of making thereof;
FIGS. 9A-9D are diagramatic representations of a three nodal embodiment of
the invention;
FIG. 10 is a diagramatic representation of a four nodal embodiment of the
invention; and
FIG. 11 is a diagram of apparatus for rotating the non-perverting mirrors
of the invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Before proceeding to an analysis of non-perverted images it is helpful to
set forth some observations regarding plane, concave and convex mirrors
and the perverted images formed thereby.
For a plane mirror, reflected wavefronts appear to come from a point as far
behind the mirror as the object being viewed is in front of the mirror.
For an extended object, the image can be calculated point by point. For
example, when two objects on either side of an observer are considered,
the image of the object on the right side is seen to the right in the
mirror, and the image of the object on the left side is seen to the left.
The same thing happens when objects above and below the mirror are
considered: the image of the upper object is seen above the image of the
lower object. Thus, in the case of a vertical mirror in an east-west
plane, when an object moves to the east (west) the image moves to the east
(west) and similarly, when an object moves up (down) the image moves up
(down). What is "reversed" in a plane mirror is forwards and backwards. An
observer and his image face each other. Thus, if an object moves to the
north, his image moves to the south. The perversion of the image is a
direct consequence of the forward-backward reversal; vertical and lateral
directions are not reversed. Any object can be oriented in space by
specifying three directions pertaining to the object: forward and
backward, left and right, and up and down. If an odd number of these
directions are reversed by an optical device, the resulting image is
perverted. A plane mirror, having only a single (forward-backward)
reversal thus produces a perverted image.
For a convex mirror, the reflected wavefronts are spheres of greater
curvature (thus smaller radius) than the original incident wavefronts. The
image appears to come from a point closer to the mirror than the
corresponding object is, and so everything appears smaller in a convex
mirror. For two objects to the right and left of an observer, the
respective images are also to the right and left, just as in a plane
mirror. The convex mirror reverses only forward and backward and so
produces perverted images.
The concave mirror is more complicated, because for different object and/or
viewer positions, the images might appear to come from in front of or in
back of the mirror. For an object inside the focus (a distance equal to
half the radius of a spherical mirror) the image appears to come from
behind the mirror, and is erect (lateral and vertical directions are not
reversed). For an object outside the focus, the image appears to come from
in front of the mirror, and is inverted (both lateral and vertical
directions are reversed). In both cases, forward and backward are
reversed, as with all mirrors. The number of reversals for erect images is
one, and for inverted images is three. Thus, since there is always an odd
number of reversals, all images in concave mirrors are perverted.
The simplest non-perverting single-reflection mirror has a concave cylinder
surface such as shown by mirror 10 in FIG. 1A. The mirror is flat in one
direction (vertical), and curved toward the observer in a direction
perpendicular to the first direction (horizontal). Images in mirror 10 can
be predicted quite well by considering mirror 10 as a combination of two
mirrors, one of which is horizontal and the other vertical. Image
formation horizontally is calculated by considering the horizontal mirror
curvature; vertical image formation is found by using the vertical mirror
curvature. The results are then combined to find the type of image formed.
This method works for any smooth mirror with similar vertical curvature
everywhere on the mirror, and also a similar horizontal curvature.
Horizontal and vertical curvatures need not be the same.
For an object that is outside the focus of mirror 10, horizontal image
formation will be similar to that in a concave mirror: right and left are
inverted. However, vertical image formation will be similar to that for a
plane mirror: up and down are not inverted. Forward and backward are
reversed, as for all mirrors, and so the total number of reversals is two
(forward-backward and right-left). As a result of the even number of
reversals, the image is not perverted. The relationship of an object to
its image in a non-perverting mirror is shown in FIG. 1B. It is noted that
the non-perverted image of a "right-hand" is a "right-hand".
When the mirror 10 is rotated by 90 degrees relative to the orientation
shown in FIG. 1A, the curvature is vertical, and the analysis is similar.
Right and left are not reversed (as in a plane mirror), whereas up and
down are reversed (as in a concave mirror). The number of reversals is
still two, so the image is non-perverted, but it appears upside down.
In the concave-cylindrical mirror, two different directions are reversed to
produce a non-perverted image. When two separate plane mirrors are used,
non-perverted images are seen as well. However, this is the result of two
successive inversions of the same direction (forward-backward), and is
thus fundamentally different from the image formation in
concave-cylindrical mirrors in that two separate reflections are required.
Any point object emits or reflects spherical wavefronts. The human eye is
adapted for spherical wavefronts, and cannot properly focus wavefronts
that are highly non-spherical. The wavefronts reflected from plane, convex
and concave mirrors are spherical, although of differing curvatures. Thus,
these waves can be easily focused by the eye. However, the wavefronts
reflected from the concave-cylindrical mirror 10 are not spherical.
Consider a nearly flat wavefront (from a distant object) that has just
reached the surface of a concave-cylindrical mirror, as indicated by 12 in
FIG. 1A. The reflected wavefronts are shown at successive time intervals
after reflection in FIGS. 2A-2C. FIG. 2A shows what happens in the
horizontal plane (parallel to the curvature) and is very similar to the
case for a concave mirror. FIG. 2B shows what happens in a vertical plane
(along the flat direction of the cylinder), and is similar to the case for
a plane mirror. The combined three-dimensional wavefronts are depicted in
FIG. 2C for the same time intervals. The waves converge on a line at t =
t.sub.3 (not a point as in the case for spherical concave mirrors), and
then assume the shape of an expanding cylinder. In order to receive waves
of the same horizontal and vertical curvature (i.e., spherical waves), the
viewer must stand at a distance far away from the mirror for the expanding
cylindrical wavefront to assume a virtually flat shape. A viewer closer
than this distance will see waves of different vertical and horizontal
curvatures, and will not be able to focus well. Poor focusing of images is
one major flaw of the concave-cylindrical mirror of the prior art.
Another major drawback of the concave-cylindrical mirror 10 is that it
provides elongated images. The amount of elongation increases as one gets
farther from the mirror. For example, if the concave radius of curvature
is labeled R, then an observer looking at himself in concave-cylindrical
mirror 10 while standing a distance of 4R from the mirror will see a
non-perverted image three times as tall as it is wide, in relation to the
original object. At a distance of 3R, the image is twice as tall as it is
wide; and at a distance of 2R, the image is just as tall as it is wide. It
is evident that the two major disadvantages of the concave-cylindrical
mirror 10 are important in different areas. A person close to the mirror
will see a relatively non-elongated image, but will not be able to focus
the wave-fronts properly because they are more asymmetrical for viewers
close to the mirror. On the other hand, when the viewer is sufficiently
far from the mirror so that image focus is acceptable the image is
extremely elongated.
One embodiment of a single-reflection non-perverting mirror of the instant
invention is shown in FIGS. 3 and 4. FIG. 3 illustrates a portion of a
torus 20 which is centered about X-Y-Z coordinate axes. The torus defines
an inner solid indicated at 30, which when cut in two or more sections
(along the Z direction) forms the mirror 40 shown enlarged in FIG. 4. The
surface of mirror 40 thus conforms to the inside surface of torus 20.
The equation for the torus is:
##EQU1##
For the portion of the torus actually used for the non-perverting mirror,
the equation is:
##EQU2##
Where --h<z<+h.
The above restrictions confine the shape of mirror 40 to an inner, "apple
core" shaped portion of the torus. The distance b will be called the
concave radius R.sub.E of the mirror, because the mirror's concave
curvature in the vertical direction is the same as that of a concave
spherical mirror of radius b. Similarly, the distance a will be called the
convex radius R.sub.X of the mirror, and denotes the curvature in the
horizontal direction. There will be no double reflections on the edges of
the mirror provided that
##EQU3##
This criterion is used because a viewer at an infinite distance will see
double reflections if h<R.sub.E /.sqroot.2; when the mirror is made so
that h = R.sub.E /.sqroot.2, the viewer at an infinite distance will just
be able to see double reflections, but closer viewers will not see any
double reflections.
In general, the wavefronts reflected from a torodial mirror assume a nearly
spherical shape closer to the mirror than do the wavefronts from a
concave-cylindrical mirror. FIG. 5A shows a nearly flat wavefront 42
approaching the surface of toroidal mirror 40. (Mirror 40 is shown rotated
90.degree. relative to the orientation of FIG. 4.) The reflected
wavefronts are shown at successive intervals in FIGS. 5B-5D. FIG. 5B shows
what happens in a horizontal plane, and is similar for the case for a
concave mirror. FIG. 5C shows what happens in a vertical plane, and is
similar to the case of a convex mirror. The combined three-dimensional
wavefronts are pictured in FIG. 5D for the same time intervals. The waves
converge to a convex curve (not a straight line, as was the case for a
concave-cylindrical mirror). The wavefronts then expand to the shape of an
outside section of a torus. Although the reflected wavefront is curved
differently in vertical and horizontal directions, the difference in
curvatures is much smaller than would be the case for a
concave-cylindrical mirror. Thus, the wavefronts assume a nearly spherical
shape much closer to the toroidal mirror 40 than for the
concave-cylinderical mirror 10. A viewer can resolve images at a distance
closer to the toroidal mirror 40 than he could for the concave-cylindrical
mirror 10.
The concave-cylindrical mirror 10 has another distinct disadvantage,
elongation of images, which also is reduced considerably in the toroidal
mirror 40. One may best illustrate image elongation mathematically. In the
following equations, the distance to the viewer's eye from the vertex
(center) of the mirror is called E; the object-vertex distance is called
P; and the mirror's radius is called R. R is negative for convex mirrors,
positive for concave mirrors, and equal to infinity for plane mirrors. For
an object of height D, the angular size of the image (.theta.) is given by
the relation
##EQU4##
FIGS. 6A-6C show the angle .theta. for similar object-viewer positions for
three different mirrors: convex, plane and concave. The ratio of the
angular image size for the curved mirror to the angular image size for a
plane mirror (P, E and D being equal to their counterparts in both cases)
is given by:
##EQU5##
After rearranging, equation (3) may be written
##EQU6##
The ratio n is negative if the curved mirror inverts the image; otherwise
it is positive.
As an example of equations (3) and (4), suppose a viewer wants to know how
far to stand from a concave mirror in order to see an inverted image of
himself that is half as big as the corresponding plane mirror image for
the same viewing distance. Then,
P = e = viewer and object distance
R = radius of concave mirror (positive)
n = -1/2 (half as big and inverted)
Inserting these parameters into equation (4) provides
P = E = 3R
thus, if the radius of the concave mirror was one foot, the viewer would
stand three feet away from it to see his inverted image half its true
height.
Next we need to compare the angular magnification of the image in both
horizontal and vertical directions. Equations (2) and (3) describe what
happens in one direction, vertical or horizontal. With V standing for
vertical and H for horizontal, the ratio of vertical image angle to
horizontal image angle is given by:
##EQU7##
The ratio M will be termed the elongation of the image. R.sub.H is the
horizontal radius of curvature, and R.sub.V is the vertical radius of
curvature. These radii need not be the same, either in magnitude or in
direction (convex or concave).
If M is positive, then the mirror either inverts images both horizontally
and vertically, or does not invert images in either of these directions.
Thus, a positive value of M indicates a perverted mirror image. If the
mirror inverts images in one direction but not the other, the image is
non-perverted, and M is negative. Thus, a perverted image that is twice as
tall as it is wide (compared to the object) would have M = +2. A
non-perverted image that is half as tall as it is wide (again compared to
the object) would have M = -(1/2). Equation (5) may be rearranged into the
useful form:
##EQU8##
This equation works for toroidal, cylindrical, spherical (concave and
convex) and plane mirrors. For instance, the requirement that M = +1
(perverted, non-elongated images) causes several cancellations in equation
2B, with the result being
R.sub.V = R.sub.H
this means that if the two curvatures are equal in magnitude and sign
(i.e., both convex, both plane, or both concave), then a perverted,
non-elongated image will be seen regardless of viewer and/or object
positions.
In the case of the prior art concave-cylindrical mirror 10, when the
curvature is horizontal (as in FIG. 1A), one may write
R.sub.h = concave radius = R.sub.E (positive)
R.sub.v = infinity (it is flat vertically). For someone viewing his own
image (P = E), equation (6) becomes after a little algebra,
##EQU9##
This equation states that elongation in concave-cylindrical mirrors is
directly proportional to viewer distance. A viewer at P = 2(R.sub.E) sees
an image for which M = 1, which defines a non-perverted image as tall as
it is wide in relationship to the object (the viewer).
A viewer at P = 3(R.sub.E) sees an image for which M = -2 (non-perverted,
and twice as tall as wide); a viewer at P = 4(R.sub.E) sees an image for
which M = -3 (non-perverted, and three times as tall as wide), and so
forth. It is evident that elongation of the image reflected from
concave-cylindrical mirror 10 increases as the viewer distance increases.
The image elongation, is, of course, a major drawback in
concave-cylindrical mirrors.
The toroidal mirror 40 (FIG. 4) produces images smaller than those of the
concave-cylindrical mirror. However, mirror 40 may be designed so that
image elongation is very slight over a wide range of viewer distances. To
show this, one uses equations (3), (4), (5) and (6), with R.sub.V =
R.sub.X = convex radius (negative) and R.sub.H = R.sub.E = concave radius
(positive). Such a mirror is shown in FIG. 7 wherein the saddle shape is
defined by arcs which are catenaries (the shape of hanging chains) of
equal magnitude but opposite direction. (Such a shape can be made with a
soap solution and a wire frame for example.) Assume, for simplicity, that
the viewer is looking at his own image, so that P = E. For the case of
R.sub.E = R = -R.sub.X (the concave radius is equal to the convex radius,
both opposite in sign), equation (5) becomes
##EQU10##
For reasonable values of viewer distance (say, P = 5R) M is about (-0.7),
i.e., the image is non-perverted and seven tenths as tall as it is wide.
As the viewer distance increases, M will approach minus one
(non-perverted, and without elongation), but will not equal minus one
until the viewer is at an infinite distance. For this reason, images
viewed in mirrors such as the saddle mirror of FIG. 7, for which the
convex radius is equal in magnitude to the concave radius, will appear
slightly elongated in the direction of the concave curvature. However, the
elongation found in mirrors of this type is considerably smaller than the
elongation found in concave-cylindrical mirrors.
Toroidal mirror 40 may be designed so that the viewer sees non-elongated
images through a specified range of distances from the mirror. For very
close viewing distances, the image will be slightly wider than it is tall,
and for more remote distances the image will be slightly taller than it is
wide. Over a reasonable range of viewing distances close to the mirror, no
elongation is apparent, and images appear much as they do in "real life".
The first step in designing such toroidal mirrors is to decide what the
magnification n is, relative to the image that would be formed in a plane
mirror, by using R.sub.E as the radius R in equation (3). From experiment,
it has been found that for someone viewing himself, a non-perverted,
non-elongated image can be resolved when the viewer is at a distance equal
to or greater than four times the concave radius of the mirror 40. Thus,
setting P = E = 4R.sub.E in equation (3), one finds that n = (-1/3). This
means the mirror must be designed so that at the normal viewing distance,
the image will be less than or equal to one-third the size of the
corresponding plane-mirror image. This represents a limitation on the
toroidal mirror; a viewer closer than 4R.sub.E will have trouble focusing
on his image. However, for the concave-cylindrical mirror, the distance
one must stand from the mirror in order to focus the image is more than
ten times the concave radius, and at this distance the image is extremely
elongated as well. Thus, although the toroidal mirror does not form
perfect images, it is a vest improvement over the prior art
concave-cylindrical mirror.
One mirror of the type of the instant invention which produces a
non-perverted, non-elongated image for a viewer at 18 inches from the
mirror, and which produces easily resolvable images, is described by:
R.sub.E = 2.52 inches = (0.14) times the viewer distance; Less than
one-fourth, as required;
R.sub.x = -3.50 inches;
h = 1.50 inches
R.sub.e /.sqroot.2 = 1.78 inches.
The value of h is less than that of R.sub.E /.sqroot.2, so there will be no
double reflections. The elongation M changes when the viewer moves toward
and away from the torodial mirror, starting from the original position (P
= E = 18 inches). Equation (6) simplifies to:
##EQU11##
when P = 12 inches, M = (-0.85). when P = 18 inches, M = (-1.00) (no
elongation) and
when P = 24 inches, M = (-1.08).
It is evident that the change in elongation is minimal for a reasonable
range of viewing positions with the toroidal mirror.
Mirror 40 is formed from a section of torus. In essence, a non-perverting
mirror in accordance with the invention has spacially coincident concave
and convex curvatures (compound curvatures). A non-perverting mirror may
thus be fabricated as illustrated in FIGS. 8A-8C. The concave and convex
radii R.sub.E and R.sub.X are those taken from the example above. FIG. 8A
shows a cylinder 50 having a half-diameter of 4.0 inches (R.sub.X = 3.5
inches and D = 0.5 inches). The cylinder is cut as shown on the right in
FIG. 8A with a cutting radius R.sub.E = 2.52 inches. Arc A-B indicates the
cutting radius. After cutting (using a lathe, for example), one obtains
the shape as shown in FIG. 8B. FIG. 8B may be cut in half (vertically
through the middle) to obtain the mirror surface of FIG. 8C. The cylinder
may be of a highly reflecting surface which may be polished to provide the
desired non-perverting mirror or may be silver coated to achieve a high
degree of reflectivity. Additionally, other methods of fabricating such
non-perverting mirrors are readily apparent such as molding and stamping
techniques.
When the compound curvature mirror or concave-cylindrical mirror is rotated
by 90.degree., the image rotates a full 180.degree. in the same direction.
On the other hand, when the mirror remains stationary and an object is
rotated 360.degree. around the mirror in a clockwise direction, the image
rotates 360.degree. in a counterclockwise direction.
A whole series of mirrors obey similar rotation laws to the compound
curvature (such as toroidal) mirror. The monkey saddle surface, shown in
FIG. 9A and heretofor a mere mathematical curiosity as regards its shape,
has interesting image rotation properties when made with a reflecting
surface to form a mirror.
This surface has an isolated parabolic point surrounded by a region of
negative curvature, which indicates a small, relatively flat central area
which grades continuously into three sets of hills and valleys
characterized by orthogonal convex and concave curvatures. A reflecting
surface as in FIG. 9A produces two non-perverted images of a given object.
The complete image is continuous everywhere except on the small,
relatively flat area at the mirror's center. One image of a viewer's body
can exchange hands with the other image of the viewer's body, as shown
diagramatically in FIGS. 9B, 9C and 9D. These figures show that an object
must go through two revolutions (720.degree.) around the stationary mirror
in order to rotate the image through one revolution (360.degree.) in the
opposite direction. As well, a rotation of the mirror by 120.degree.
rotates the image of a stationary object through 180.degree. in the same
direction.
A surface with four sets of hills and valleys (four "nodes") produces three
non-perverted images. Such a surface is shown in FIG. 10. A rotation of
this "nodal" mirror by 135.degree. rotates the images by 180.degree.. When
the mirror is stationary, three revolutions of the object around the
mirror are required for one revolution of the image.
In general, if the surface has m nodes, there will be (m - 1) non-perverted
images. To produce an image rotation of X degrees in a particular
direction, the mirror must be rotated through
##EQU12##
degrees in the same direction.
When the mirror is stationary, an image rotation of Y degrees in one
direction is caused by a rotation of the object by (m - 1)Y degrees in the
opposite direction.
For a two-node mirror (see FIG. 7), m is 2. Thus the number of images will
be (m - 1) = 2 - 1 = 1. To rotate the image through 180.degree., the
mirror must be rotated through
##EQU13##
And to rotate the image through 360.degree. when the mirror is stationary,
the object must be rotated through
(m - 1) (y) = (2 - 1) (360.degree.) = 360.degree..
The toroidal mirror 40 of FIG. 4 as well as the compound mirrors shown in
FIG. 8C are both essentially two-nodal mirrors in that they have a single
pair of intersecting convex and concave surface regions.
FIG. 11 shows a means for rotating mirrors such as any of the
non-perverting mirrors of the invention. Motor 50 is connected to a
holding means 60 to rotate same. The mechanical connection between motor
50 and holding means 60 is indicated by a dotted line. Holding means 60
has secured thereto (via adhesive means for example) a non-perverting
compound curvature mirror (such as mirror 40) in accordance with the
invention. Image and object arrows are depicted, and it is apparent that
upon rotation of the holding means 60 and mirror the image will rotate in
accordance with the rotation laws set forth herein.
The two-nodal mirror of FIG. 7 is simply a special case (R.sub.E = R.sub.X)
of the toroidal mirror 40 of FIGS. 4 and 8C.
Although the invention has been described with particular reference to the
preferred embodiment, it is understood that certain modifications and
improvements may be evident to those skilled in the art and the invention
is intended to cover all modifications which do not depart from the spirit
and scope of the invention.
* * * * *
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