HUMAN AND ANIMAL VISION

In the context of human vision, the term “field of view” is typically used in the sense of a restriction to what is visible by external

apparatus, like spectacles[2] or virtual reality goggles. Note that eye movements do not change the field of view.

If the analogy of the eye’s retina working as a sensor is drawn upon, the corresponding concept in human (and much of animal

vision) is the visual field.[3] It is defined as “the number of degrees of visual angle during stable fixation of the eyes”. [4] Note that eye

movements are excluded in the definition. Different animals have different visual fields, depending, among others, on the placement

of the eyes.Humans have an almost 180-degree forward-facing horizontal diameter of their visual field, while some birds have a

complete or nearly complete 360-degree visual field. The vertical range of the visual field in humans is typically around 135 degrees.

The range of visual abilities is not uniform across the visual field, and varies from animal to animal. For example, binocular vision,

which is the basis for stereopsis and is important for depth perception, covers only 114 degrees (horizontally) of the visual field in

humans;[5] the remaining peripheral 60–70 degrees have no binocular vision (because only one eye can see those parts of the visual

field). Some birds have a scant 10 or 20 degrees of binocular vision.

Similarly, color vision and the ability to perceive shape and motion vary across the visual field; in humans the former is concentrated

in the center of the visual field, while the latter tends to be much stronger in the periphery. The physiological basis for that is the

much higher concentration of color-sensitive cone cells and color-sensitive parvocellular retinal ganglion cells in the fovea – the

central region of the retina – in comparison to the higher concentration of color-insensitive rod cells and motion-

sensitive magnocellular retinal ganglion cells in the visual periphery. Since cone cells require considerably brighter light sources to

be activated, the result of this distribution is further that peripheral vision is much more sensitive at night relative to foveal vision. [3]

Most geometry so far has involved triangles and quadrilaterals, which are formed by

intervals on lines, and we turn now to the geometry of circles. Lines and circles are the

most elementary figures of geometry − a line is the locus of a point moving in a

constant direction, and a circle is the locus of a point moving at a constant distance

from some fixed point − and all our constructions are done by drawing lines with a

straight edge and circles with compasses. Tangents are introduced in this module, and

later tangents become the basis of differentiation in calculus.

The theorems of circle geometry are not intuitively obvious to the student, in fact most

people are quite surprised by the results when they first see them. They clearly need to

be proven carefully, and the cleverness of the methods of proof developed in earlier

modules is clearly displayed in this module. The logic becomes more involved − division

into cases is often required, and results from different parts of previous geometry

modules are often brought together within the one proof. Students traditionally learn a

greater respect and appreciation of the methods of mathematics from their study of this

imaginative geometric material.

The theoretical importance of circles is reflected in the amazing number and variety of

situations in science where circles are used to model physical phenomena. Circles are

the first approximation to the orbits of planets and of their moons, to the movement of

electrons in an atom, to the motion of a vehicle around a curve in the road, and to the

shapes of cyclones and galaxies. Spheres and cylinders are the first approximation of

the shape of planets and stars, of the trunks of trees, of an exploding fireball, and of a

drop of water, and of manufactured objects such as wires, pipes, ball-bearings, balloons,

pies and wheels.

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CONTENT

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RADII AND CHORDS

We begin by recapitulating the definition of a circle and the terminology used for circles.

Throughout this module, all geometry is assumed to be within a fixed plane.

A circle is the set of all points in the plane that are a fixed distance (the radius)

from a fixed point (the centre).

Any interval joining a point on the circle to the

centre is called a radius. By the definition of a circle, any two radii have the

same length. Notice that the word ‘radius’ is being used to refer both to these

intervals and to the common length of these intervals.

An interval joining two points on the circle is called achord.

A chord that passes through the centre is called a diameter. Since a diameter

consists of two radii joined at their endpoints, every diameter has length equal to

twice the radius. The word ‘diameter’ is use to refer both to these intervals and

to their common length.

A line that cuts a circle at two distinct points is called a secant. Thus a chord is

the interval that the circle cuts off a secant, and a diameter is the interval cut off

by a secant passing through the centre of a circle centre.

Symmetries of a circle

Circles have an abundance of symmetries:

A circle has every possible rotation symmetry about its

centre, in that every rotation of the circle about its 

centre rotates the circle onto itself.

 

 

If AOB is a diameter of a circle with centre O, then the 

reflection in the line AOB reflects the circle onto itself. 

Thus every diameter of the circle is an axis of symmetry.

As a result of these symmetries, any point P on a circle 

can be moved to any other point Q on the circle. This can 

be done by a rotation through the angle θ =  POQ about 

the centre. It can also be done by a reflection in the diameter 

AOB bisecting  POQ. Thus every point on a circle is essentially 

the same as every other point on the circle − no other figure in 

the plane has this property except for lines.