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In mathematics, a projective line is speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases. There are many equivalent ways; this definition is a special instance of the general definition of a projective space. An arbitrary point in the projective line P1 may be represented by an equivalence class of homogeneous coordinates, which take the form of a pair of elements of K that are not both zero. Two such pairs are equivalent if they differ by an overall nonzero factor λ: ∼; the projective line may be identified with the line K extended by a point at infinity. More the line K may be identified with the subset of P1 given by; this subset covers all points in P1 except one, called the point at infinity: ∞ =. This allows to extend the arithmetic on K to P1 by the formulas 1 0 = ∞, 1 ∞ = 0, x ⋅ ∞ = ∞ if x ≠ 0 x + ∞ = ∞ if x ≠ ∞ Translating this arithmetic in terms of homogeneous coordinates gives, when does not occur: + =, ⋅ =, − 1 =.

The projective line over the real numbers is called the real projective line. It may be thought of as the line K together with an idealised point at infinity ∞. An example is obtained by projecting points in R2 onto the unit circle and identifying diametrically opposite points. In terms of group theory we can take the quotient by the subgroup. Compare the extended real number line, which distinguishes ∞ and −∞. Adding a point at infinity to the complex plane results in a space, topologically a sphere. Hence the complex projective line is known as the Riemann sphere, it is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface. The projective line over a finite field Fq of q elements has q + 1 points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates, q of these points have the form: for each a in Fq,and the remaining point at infinity may be represented as.

Quite the group of homographies with coefficients in K acts on the projective line P1. This group action is transitive, so that P1 is a homogeneous space for the group written PGL2 to emphasise the projective nature of these transformations. Transitivity says that there exists a homography that will transform any point Q to any other point R; the point at infinity on P1 is therefore an artifact of choice of coordinates: homogeneous coordinates ∼ express a one-dimensional subspace by a single non-zero point lying in it, but the symmetries of the projective line can move the point ∞ = to any other, it is in no way distinguished. Mu

Willem Frederick August Winckel was a Dutch military officer and pilot who flew for the Royal Netherlands East Indies Air Force in World War II. During the attack on Broome, Western Australia, on 3 March 1942, Winckel managed to land his plane full of refugees safely on the Broome airstrip just before the Japanese attack, he dismounted the plane's machine gun and shot down one of the Japanese fighters, the only Allied "kill" during the attack. Shortly after the attack on Broome, Winckel was sent on a mission to Bandung, Java, to recover several officers from the Allied headquarters, under threat of being overrun, he served the remainder of the war with No. 18 Squadron RAAF in Australia. After World War II, he fought in the Dutch Politionele acties. Winckel was born on 3 November 1912 in Muntok, Bangka Island, in what was the Dutch East Indies. On the island his parents ran a trading store in commodities such as rubber and sugar. Winckel had an untroubled childhood and at age fifteen was sent back to the Netherlands to study at the nautical school in Delfzijl.

He stayed with his grandparents while studying, three years became a seaman first class for the merchant marine. He specialised in navigation. In 1931, when he was aged nineteen, Winckel started working as a quartermaster on the freighter Pallau, which sailed between the Netherlands and Java. After his work on the ship he went to the island of Borneo to work on oil rigs. By 1935 he returned to the Netherlands once more, this time to work for the Royal Netherlands Navy as a submarine torpedo calibrator. During his time in the Navy, a friend showed him a flyer for a pilot position at the Netherlands East Indies Air Force, being formed at the time. Winckel joined the Air Force in 1935 and became a pilot at age 23. Between 1935 and 1942 he flew transports across the Dutch East Indies in Lockheed Lodestars. On 3 March 1942, Flight Lieutenant Winckel was transporting refugees from the Bandung, Dutch East Indies, to Perth, Western Australia, in his Lockheed Lodestar, he had just stopped on the Broome Airfield to refuel when Broome was attacked by nine Japanese Zero fighter planes on a long-distance raid from Kupang, Timor.

Winckel saw the danger of his plane being on the open airstrip and proceeded to dismount the Colt 7.99mm machine gun from the plane. When one of the Zeros, piloted by Osamu Kudō, flew over at low altitude, Winckel managed to shoot him out of the air. Winckel was thereby credited with the raid's only "kill" on the Allied side, he believed that he hit another Japanese plane that had to ditch in the sea. The Japanese destroyed 22 Allied aircraft at Broome, many of them flying boats used for refugee transport and moored in Roebuck Bay. Winckel suffered severe burns to his hand during the attack, as he had to hold the machine-gun barrel to aim, his hand was treated and Winckel flew medical evacuations for the next two days without rest. Meanwhile, Japanese forces were threatening the Allied headquarters at Bandung. On 5 March, the Allied command decided that several senior RAF and RAAF officers had to be evacuated from Bandung. Winckel was chosen as the pilot, he protested the decision, arguing that his lack of rest would hinder his chances of finding Bandung in the dark as he did not have any electronic aids.

However, Winckel was seen to be one of the most experienced pilots because of his terrain knowledge as a flight instructor at Bandung and his recent refugee flights. Another factor was that his Lodestar, although it had been damaged in the attack on Broome, was considered the most suitable plane for the evacuation mission. Winckel flew to Java, he picked up fourteen passengers, refuelled the plane and returned to Australia. Following the attack on Broome and his evacuation mission, Winckel was sent to Moruya, New South Wales, where he joined No. 18 Squadron RAAF, formed on 4 April 1942. From Moruya he patrolled the Eastern Australian shore. At the end of May 1942, a Japanese midget submarine attacked Sydney Harbour. Winckel and his fellow pilots were subsequently ordered to patrol for submarines. During a patrol on 5 June 1942, Winckel attacked what he believed was a Japanese submarine in his B-25 Mitchell bomber; the perceived success was welcomed after the recent submarine attack and Winckel was congratulated by Australian Prime Minister John Curtin.

This was one of a large number of claimed successes against Japanese submarines during this period. However, postwar research indicates that no Japanese submarines were sunk off the Australian east coast during the war. Winckel spent the rest of the war with No. 18 Squadron and served from MacDonald Airfield and Batchelor Airfield in Northern Australia as a pilot on operations, as an instructor working in Canberra. During the war, he was nicknamed "Lucky Bill" by his colleagues and "Babyface Killer" by the Japanese. While a member of No. 18 Squadron, Winckel flew to the Dutch East Indies in late 1945 as part of an unauthorized "Recovery of Allied Prisoners of War and Internees" mission. Winckel was unable to find them; some time another pilot flew over the Banjoe Biroe internment camp and saw the name'NINI' written on the ground with flour. This was the name of Winckel's sister, to draw the attention of her brother, who she assumed was looking for her. Shortly afterwards, Winckel himself flew over the camp and dropped a letter to her, followed four days by a supply of food.

After the war, Winckel was sent back to Indonesia to fight in the Dutch Politionele acties. Winckel had met his future wife Yvonne prior to Wo

Tommaso Guzzoni, C. O. was a Roman Catholic prelate. Tommaso Guzzoni was born in Benevento, Italy on 28 September 1632. On 13 May 1653, he professed as a member of the Congregation of the Oratory of Saint Philip Neri and was ordained a deacon on 19 December 1654 and ordained a priest on 18 September 1655. On 13 January 1681, he was appointed during the papacy of Pope Innocent XI as Bishop of Sora. On 26 January 1681, he was consecrated bishop by Alessandro Crescenzi, Bishop of Recanati e Loreto, with Pier Antonio Capobianco, Bishop Emeritus of Lacedonia, Antonio Savo de' Panicoli, Bishop of Termoli, serving as co-consecrators, he served as Bishop of Sora until his resignation on 5 December 1702. He died on 8 November 1704. While bishop, he was the principal co-consecrator of: Cheney, David M. "Diocese of Sora-Cassino-Aquino-Portecorvino". Catholic-Hierarchy.org. Retrieved June 16, 2018. Chow, Gabriel. "Diocese of Sora-Cassino-Aquino-Portecorvino". GCatholic.org. Retrieved June 16, 2018

Claire Isabelle Geo Pommet, known professionally as Pomme, is a French singer and musician. Claire Pommet grew up in the Lyon area, she learned music theory from the age of 6, joined a children's choir, La Cigale de Lyon, at age 7 and learned to play the cello at age 8. Her mother and believer, plays the flute, she publishes videos on the YouTube web platform. In September 2017, at the age of 21, she performed for the first time four times at La Boule noire in Paris. In October she released her first chanson française album, titled À peu près; the journalists of Libération and Le Figaro emphasize the quality of her live performances, during which she plays the automatic harp and the guitar. Pomme performed in the first part of the Asaf Avidan tour in autumn 2017. In February 2018, she performed at the Café de la Danse in Paris, after having performed as an opening act for Louane and Vianney took the stage in La Cigale in mid-2018 and in La Trianon in early 2019; the lyrics she writes and composes and half of the repertoire she interprets evoke themes related to love, death and "everyday situations that resort to romanticism".

Love, in her lyrics, is not only heterosexual, but bisexual or homosexual. She explains to Télérama: "I am comfortable with my homosexuality, for example, using female pronouns in my songs, and I think. As a teenager, I would have liked to recognize myself in lesbian singers". In 2020, she wins the Victoires de la Musique prize for "revelation album" of the year with her opus "les failles". Director information obtained from each video description on YouTube. Official website

Santa Caterina d'Alessandria is a Gothic-style, Roman Catholic church in Pisa, region of Tuscany, Italy. It is mentioned for the first time in 1211 associated with a hospital; the current edifice was built between 1251 and 1300, commissioned by Saint Dominic himself, entrusted to the friars of his order. The façade has a pointed shape with white and grey marble, with, in the upper section, two order of small Gothic loggias and a central rose window; the interior, after a fire in 1651, is on a single large hall. Renovated in the 18th century, it houses works by Lippo Memmi, Fra Bartolomeo, Santi di Tito, Aurelio Lomi, Raffaello Vanni, Pietro Dandini and marble sculptures by Andrea Pisano and his son Nino Pisano. Notable is the tomb of Gherardo Compagni, decorated with a late 16th-century "Pietà" statue; the wooden pulpit from the 17th century, according to the tradition, was that from which St Thomas Acquinas preached. In 1320, Simone Martini executed for this church the Saint Catherine of Alexandria Polyptych, one of his best known works.

The painting has been moved to the San Matteo Museum in Pisa. The church is flanked by a bell tower with mullioned windows, attributed to Giovanni di Simone. Barsali, U.. Castelli. Storia e Capolavori di Pisa. Florence: Bonechi. Donati, Roberto. Pisa. Arte e storia. Narni: Plurigraf. Page about the church