Catoptrics

Light path of a Newtonian (catoptric) telescope

Catoptrics (from katoptrikós, "specular", from katoptron "mirror" ) deals with the phenomena of reflected light and image-forming optical systems using mirrors.

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Chromatic aberration

Failure of a lens to focus all colors to the same point.

Photographic example showing high quality lens (top) compared to lower quality model exhibiting transverse chromatic aberration (seen as a blur and a rainbow edge in areas of contrast.)
Chromatic correction of visible and near infrared wavelengths. Horizontal axis shows degree of aberration, 0 is no aberration. Lenses: 1: simple, 2: achromatic doublet, 3: apochromatic and 4: superachromat.
Color shifting through corner of eyeglasses.
Severe purple fringing can be seen at the edges of the horse's forelock, mane, and ear.
This photo taken with the lens aperture wide open resulting in a narrow depth-of-field and strong axial CA. The pendant has purple fringing in the near out-of-focus area and green fringing in the distance. Taken with a Nikon D7000 camera and an AF-S Nikkor 50mm f/1.8G lens.
Severe chromatic aberration

Modern telescopes, as well as other catoptric and catadioptric systems, continue to use mirrors, which have no chromatic aberration.

Dioptrics

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Dioptrics is the branch of optics dealing with refraction, similarly the branch dealing with mirrors is known as catoptrics.

Euclid

Not to be confused with Euclid of Megara.

Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid
Euclidis quae supersunt omnia (1704)
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid's construction of a regular dodecahedron.
Construction of a dodecahedron by placing faces on the edges of a cube.
19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History

Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J O'Connor and E F Robertson who name Theon of Alexandria as a more likely author.

Ibn al-Haytham

Arab mathematician, astronomer, and physicist of the Islamic Golden Age.

Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen's Book of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
The structure of the human eye according to Ibn al-Haytham. Note the depiction of the optic chiasm. —Manuscript copy of his Kitāb al-Manāẓir (MS Fatih 3212, vol. 1, fol. 81b, Süleymaniye Mosque Library, Istanbul)
The theorem of Ibn Haytham
Hevelius's Selenographia, showing Alhasen [sic] representing reason, and Galileo representing the senses.
Alhazen's geometrically proven summation formula
The lunes of Alhazen. The two blue lunes together have the same area as the green right triangle.
Cover page of the Latin translation of Kitāb al-Manāẓir

His work on catoptrics in Book V of the Book of Optics contains a discussion of what is now known as Alhazen's problem, first formulated by Ptolemy in 150 AD. It comprises drawing lines from two points in the plane of a circle meeting at a point on the circumference and making equal angles with the normal at that point.

Theon of Alexandria

Greek scholar and mathematician who lived in Alexandria, Egypt.

Proto-Greek area of settlement (2200/2100-1900 B.C.) suggested by Katona (2000), Sakelariou (2016, 1980, 1975) and Phylaktopoulos (1975)

Catoptrics. The authorship of this treatise, ascribed to Euclid, is disputed. It has been argued that Theon wrote or compiled it. The Catoptrics concerns the reflection of light and the formation of images by mirrors.

Latin translations of the 12th century

Latin translations of the 12th century were spurred by a major search by European scholars for new learning unavailable in western Europe at the time; their search led them to areas of southern Europe, particularly in central Spain and Sicily, which recently had come under Christian rule following their reconquest in the late 11th century.

Albohali's De Iudiciis Natiuitatum was translated into Latin by Plato of Tivoli in 1136, and again by John of Seville in 1153. Here is the Nuremberg edition of John of Seville's translation, 1546.
Ibn Butlan's Tacuinum sanitatis, Rhineland, 2nd half of the 15th century.
King Alfonso X (the Wise)
Al-Razi's Recueil des traités de médecine translated by Gerard of Cremona, second half of the 13th century.
Depiction of the Persian physician Al-Razi, in Gerard of Cremona's "Recueil des traités de medecine" 1250–1260.

Optica and Catoptrica: from Greek, probably Sicily

Philosophy of science

Branch of philosophy concerned with the foundations, methods, and implications of science.

Karl Popper in the 1980s
The expectations chickens might form about farmer behavior illustrate the "problem of induction."
Seen through a telescope, the Einstein cross seems to provide evidence for five different objects, but this observation is theory-laden. If we assume the theory of general relativity, the image only provides evidence for two objects.
Francis Bacon's statue at Gray's Inn, South Square, London
For Kuhn, the addition of epicycles in Ptolemaic astronomy was "normal science" within a paradigm, whereas the Copernican revolution was a paradigm shift.
Jeremiah Horrocks makes the first observation of the transit of Venus in 1639, as imagined by the artist W. R. Lavender in 1903
Paul Karl Feyerabend
Hegel with his Berlin students
Sketch by Franz Kugler
Peter Godfrey-Smith was awarded the Lakatos Award for his 2009 book Darwinian Populations and Natural Selection, which discusses the philosophical foundations of the theory of evolution.
A fragment of the Hippocratic Oath from the third century.
Wilhelm Wundt (seated) with colleagues in his psychological laboratory, the first of its kind.

The eleventh century Arab polymath Ibn al-Haytham (known in Latin as Alhazen) conducted his research in optics by way of controlled experimental testing and applied geometry, especially in his investigations into the images resulting from the reflection and refraction of light.

Fermat's principle

Link between ray optics and wave optics.

Fig.1:Fermat's principle in the case of refraction of light at a flat surface between (say) air and water. Given an object-point A in the air, and an observation point B in the water, the refraction point P is that which minimizes the time taken by the light to travel the path APB. If we seek the required value of x, we find that the angles α and β satisfy Snell's law.
Fig.2:Two points P and P′ on a path from A to B. For the purposes of Fermat's principle, the propagation time from P to P′ is taken as for a point-source at P, not (e.g.) for an arbitrary wavefront W passing through P. The surface Σ (with unit normal n̂ at P′) is the locus of points that a disturbance at P can reach in the same time that it takes to reach P′; in other words, Σ is the secondary wavefront with radius PP′. (The medium is not assumed to be homogeneous or isotropic.)
Fig.3:An experiment demonstrating refraction (and partial reflection) of rays — approximated by, or contained in, narrow beams
Fig.4:Two iterations of Huygens' construction. In the first iteration, the later wavefront W′ is derived from the earlier wavefront W by taking the envelope of all the secondary wavefronts (gray arcs) expanding in a given time from all the points (e.g., P) on W. The arrows show the ray directions.
Pierre de Fermat (1607–1665)
Christiaan Huygens (1629–1695)
Pierre-Simon Laplace (1749–1827)
Thomas Young (1773–1829)
Augustin-Jean Fresnel (1788–1827)

Hero of Alexandria, in his Catoptrics (1st century CE), showed that the ordinary law of reflection off a plane surface follows from the premise that the total length of the ray path is a minimum.

Fresnel lens

Type of composite compact lens developed by the French physicist Augustin-Jean Fresnel for use in lighthouses.

First-order rotating catadioptric Fresnel lens, dated 1870, displayed at the Musée national de la Marine, Paris. In this case the dioptric prisms (inside the bronze rings) and catadioptric prisms (outside) are arranged to concentrate the light from the central lamp into four revolving beams, seen by sailors as four flashes per revolution. The assembly stands 2.54 metres tall and weighs about 1.5 tonnes.
1: Cross-section of Buffon/Fresnel lens. 2: Cross-section of conventional plano-convex lens of equivalent power. (Buffon's version was biconvex. )
Close-up view of a flat Fresnel lens shows concentric circles on the surface
Makapuu Point Light
Cape Meares Lighthouse; first-order Fresnel lens
A plastic Fresnel lens sold as a TV-screen enlarging device
The Fresnel lens used in the Sinclair FTV1 portable CRT TV, which enlarges the vertical aspect of the display only
Inchkeith lighthouse lens and drive mechanism
Optical landing system on US Navy aircraft carrier USS Dwight D. Eisenhower
Cross-section of a first-generation Fresnel lighthouse lens, with sloping mirrors m,n above and below the refractive panel RC (with central segment A). The design was later improved by replacing the mirrors with reflective prisms to reduce losses. If the cross-section in every vertical plane through the lamp L is the same (cylindrical symmetry), the light is spread evenly around the horizon.
First-order group-flashing Fresnel lens, on display at the Point Arena Lighthouse Museum, Point Arena Lighthouse, Mendocino County, California. The three dioptric panels (inside the brass rings) and three catadioptric panels (outside) are partly split in two, giving three double-flashes per rotation.
First-order lens
Close-up of a second-order lens
Third-order lens (St. Simons Island Light)
Fourth-order lens (Sekizaki Lighthouse, Oita, Japan)
Fifth-order lens (Jones Point Light)
Sixth-order lens (Ponce de Leon Inlet Light)

If this was supplemented by reflecting (catoptric) rings above and below the refracting (dioptric) parts, the entire apparatus would look like a beehive.

Figurative system of human knowledge

Tree developed to represent the structure of knowledge itself, produced for the Encyclopédie by Jean le Rond d'Alembert and Denis Diderot.

Classification chart with the original "figurative system of human knowledge" tree, in French.

Catoptrics.