Inversion Preserves Angles Between Lines
- In this article, we explore the concept of inversion and how it preserves angles between lines.
- We discuss the different scenarios when lines are inverted in a circle with center O.
- We show that if the center of inversion is on one of the lines, the resulting figures have the same angle as the original lines.
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8.4.1 Inversion Preserves Angles Between Lines Suppose AB and CD are two lines in the plane that intersect at the point P. If they are inverted in a circle with center O, we will show that the resulting gures make the same angle with each other. The easiest case is if O, the center of inversion, happens to be the same as P. Since both lines pass through the center of inversion, they are transformed into themselves so their images trivially have the same angle between them. If the center of inversion is not on one of the lines (but it may be on the other), then the line that does not contain the center will be inverted to a circle passing throughO. Suppose O is not on AB. Consider the line OX passing throughO that is parallel to AB . Under inversion, OX is transformed transformed into itself (since it passes through the center of inversion), and AB is inverted into a circle K passing through O. Then OX must be tangent to K, since it touches K at O, and if it intersected K in more than one place, OX and AB would intersect, which is impossible since they are parallel. If O is on CD, then CD is inverted to itself, and its angle with OX is clearly unchanged (OX is also inverted into itself). But CD makes the same angle with OX as with AB since they are parallel, so the inverses of AB and CD make the same angle. Finally, if O is on neither line, then both AB and CD are inverted to circles that meet at O, and the lines tangent to those circles at O (OX and OY in the gure) are parallel to the original lines, so they make the same angles as did the original lines.
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「8.4.1 相反は、交差する線の成す角度を保存する 線分ABと線分CDが点Pで交わるとする。Oを中心とする円内でそれらが相反するなら、得られた図形が互い同じ角度になることを示そう。 最も単純な場合を考える。相反の中心点である点Oが点Pと重なるとする。 二つの線分が相反の中心点を通るので、二つの線分はそれら自身に変換され、単純に互いに成す角度は同じになる。 相反の中心点が一方の線分上にない場合（ただし、もう一方の線分上にあってもよい）、中心点を含まない線分は、点Oを通る円に変換される。Oが線分AB上にないとする。点Oを通り線分ABに平行な線分OXを考える。相反すると、線分OXはそれ自身に変換され（なぜなら、相反の中心点を通るので）、線分ABは点Oを通る円Kに変換される。 すると線分OXは円Kに接線でなければならない。なぜなら、それは点Oにおいて円Kに接しており、もし円Kに2箇所以上で交わるならば、線分OXとABも交わり、それらが平行ということに反するからである。 点Oが線分CD上にあるなら、線分CDはそれ自身に変換され、それが線分OXと成す角度は明らかに不変である（線分OXもまたそれ自身に変換される）。しかし、線分CDは線分ABと平行であるので、線分ABが線分OXと成す角度と同じ角度を線分OXに対し成す。故に、線分ABと線分CDの反転は、同じ角度を保持する。 最後に、点Oが線分ABと線分CDのどちらの線分上にもない場合、線分ABと線分CDは、点Oで接するような二つの円に変換される。点Oにおいてそれらの円に接する線分（OXとOYは図を参照）は、元の線分と平行である。故に、それらの線分が成す角度は、元の線分が成す角度と等しい。」 ということらしいです。gureはfigureのtypoと解釈しました。変換は射影かもしれません。後は数学カテで確認してください。
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8.4.3 Orthogonal Circles An interesting situation to consider is when circles (or lines, or lines and circles) intersect at 90. In any of these cases, the two figures are called orthogonal. In figure 8.9 the circle in the center is surrounded by a set of circles that are orthogonal to it. Since angles are preserved under inversion, what happens if a circle is inverted through another circle to which it is orthogonal? The answer is that the circle is inverted into itself. To see why, remember that the circle of inversion is unchanged by the inversion. The circle orthogonal to it is mapped to a circle that meets it at the same points, and remains orthogonal. Therefore it must be mapped to the same circle. (Note that this does not mean that the points are not moved by the inversionthe ones outside are mapped to points inside and vice-versa, but every point on that circle is inverted to another point on the same circle.)
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8.4 Preservation of Angles Under Inversion When we say that inversion in a circle preserves angles, exactly what is meant? It is clear what is meant by the angle between two lines, but what is the definition of the angle between a line and a circle, or the angle between two circles? The sensible way to define the angle between curves (where a straight line is a special type of curve) is to say that the angle between them is the same as the angle between their tangent lines at the point of intersection. Thus if a pair of circles intersect at a point P and you want to measure the angle between them, construct the tangents to each of the circles at P and measure the angle between those tangent lines. If you have a circle and a line, just nd the tangent to the circle at the point of intersection and measure its angle relative to the line.
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8.3.5 Inversion with Straightedge and Compass All the results above are theoretical, and can be very useful in constructing a proof, but if you actually have to draw the inverse of a line or a circle, how do you do it? If the result is a line, simply select two points on the line or circle that is to be inverted and invert them as described in Section 8.2. Then connect those two points with a line. If the result is a circle, simply invert three different points, and you will be left with three points that must lie on the required circle. The intersection of the perpendicular bisectors of the segments connecting the points is the center of the required circle. Of course if you are using Geometer, just use the built-in inversion commands: VC=>V Inv, LC=>C Inv and CC=>C Inv. Since both lines and circles usually go to circles, Geometer assumes that the result is always a circle; if the result is a line, what is drawn is effectively an innitely large circle.
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With that kind of money at stake, it is hardly surprising that industrial-scale technology has caught on. Fishing hooks are on “long-lines” sometimes more than 50 miles in length. Factory ships that can hold 1000 tons of fish store and process the catches. Fishing on such a massive scale can exhaust a fishing ground, but when that happens, the factory ships just move on. The bottom of what is known as the continental slope, between 600 and 6000 feet deep, is home to several species that swim in schools and grow as long as two to three feet. Their presence opened up the continental slope to industrial deep-sea fishing that pays off very well. The usual method, known as bottom trawling, is to drag a huge net, with 15 tons of weight, across the ocean floor, catching everything in its path. What are the environmental costs? No one really knows ------ and that is part of the problem. According to a recent issue of Natural History, catch limits for deep-sea fishes were set “essentially by guessing, relying on knowledge of shallow-water species. They took no account of the far slower birth rate in a typical population of deep-sea fishes.” Two deep-sea species have already been reduced: the orange roughy and the Chilean sea bass. ----------------------------------------------------------- たぶん漁業の話なのですが、最初の方の釣り鉤は～の文と、途中の法律、警察の仕事、厳しい刑罰～あたりの文はなんとなく意味がつかめたのですがそれ以外はさっぱりです・・・ 大まかな流れだけでもいいので教えてくださると嬉しいです
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和訳をよろしくお願いします！ On November 15 a festival is held for children seven （siti）five(go) and three(san) years old, and so the festival is called shichi-go-san. The reason why we have a celebration for these years is because odd numbers are believed to be lucky since they cannot be divided by two. In addition, we are told that since the ages of 7, 5 and 3 years are important growth steps for children, it is, therefore, significant to honor the children of these ages. In many parts of Japan, a child is received as a member of a shrine parts when he or she is seven. Though the are variations according to locality, in general, boys who are three and five and girls who are three and seven are dressed in their best clothes, and are taken by their parents to large and prestigious shrines, and very often A to the shrines of their local deities. It is customary after the visit to buy chitose-ame(thousand-year candy), sold at the stalls on the grounds of the shrine. よろしくお願いしますm(_ _)m
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この文章の和訳を教えてください。 My wanderings in life has always told me that sterotype behaviour is learned and fickle...the real person is always hiding within too scared to appear in fear of rejection....and we all have those same fears...so perhaps we can all sterotype our fears as well....others lay in ignorance to their real self....probably because they are fightened of who they are and they mask the truth of themselves... 拒否されることに対する恐れから本当の人格を隠そうとしてしまうっていうことなんでしょうか？ 単語から大体内容はつかめてる気はするのですが、細かく読み取れません。 どなたか教えてくださるとうれしいです。
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Eye movement is another reliable indication of lying. It is normal for a person to look up to the left when thinking about the (1) and up to the right when thinking about the (2). For example, if you ask people the color of their first car, or some other question from their past, they will look up to their left to access their memory. If instead they look up to their right, they are thinking of the future, which indicates that they are likely to be making up a response. Policemen are trained to routinely monitor eye movement during interviews. According to Paul Ekman, a professor of psychology at the University of California, San Francisco, there are small body movements called "micro-gestures " that are useful to observe. Two of the most common micro-gestures that are associated with lies are the nose wrinkle and the mouth crul. The nose wrinkle is the same gesture that occurs naturally when you smell something unpleasant. The other facial micro-gesture is a slight downward curl of the corners of the mouth. Even liars who make a conscious effort to hide all of thier major body gestures will still transmit micro-gestures. People sometimes lie, but their body language always tells the truth! ・(1)に入る単語を文中から抜き出してください。 ・(2)に入る単語を文中から抜き出してください。
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Were the observer, whether casual or skilled, to reside for some length of time in an asylum, and thus to make himself practically acquainted with the ways, thoughts, and feelings of its inmates, he would certainly discover how great a mistake it is to suppose, as is often done, that they are always so alienated from themselves and from their kind as not to be influenced by the same motives as sane persons [are influenced] in what they do or forbear to do. When an insane person is on his trial for some criminal offence, it is commonly taken for granted by the lawyers that if an ordinary motive for the act, such as anger, revenge, jealousy, or any other passion, can be discovered, there is no ground to allege insanity, or, at any rate, no ground to allege exemption from responsibility by reason of insanity. The ideal madman whom the law creates is supposed to act without motives, or from such motives at it enters not into the mind of a sane person to conceive; and if some one, who is plainly mad to all the world, acts from an ordinary motive in the perpetration of an offence, he is presumed to have acted sanely and with full capacity of responsibility. No greater mistake could well be made. Much of the success of the modern humane treatment of insanity rests upon the recognition of two principles: first, that the insane have like passions with those who are not insane, and are restrained from doing wrong, and constrained to do right, by the same motives which have the same effects in sane persons; secondly, that these motives are only effective within limits, and that beyond these limits they become powerless, the hope of reward being of no avail, and the expectation or infliction of punishment actually provoking more unreason and violence.
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