Preservation of Angles Under Inversion
- Understanding the concept of angle preservation under inversion in a circle
- Defining the angle between a line and a circle, or between two circles
- Measuring angles between curves using tangent lines at points of intersection
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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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「8.4 相反における角度の保存 円内での相反が角度を保存するとは、どういう意味だろうか？ 二つの線分の角度という意味は明らかであるが、線分と円が成す角度、あるいは二つの円の成す角度とはなんだろうか？ 曲線における角度（直線が成す角度は曲線の場合の特殊な例である）を適切に定義するには、二つの曲線が成す角度が、その交わる点におけるそれぞれの曲線の接線が成す角度とすることである。このようにして、二つの円が点Pで交わっており、その成す角度を決定したいなら、点Pにおけるそれぞれの円の接線を作り、その接線が成す角度を測ればよいのである。円と直線ならば、交点における円の接線を引き、その接線と直線との角度を測ればよいのである。」 らしいです。後は数学カテで確認してください。
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Preservation of Angles Under Inversionの説明の一部と思いますが、 掲載の文章の要旨では判然としない所があります。内容は以下の通りです。 ２本の直線が交わるときにその間に角度ができる、しかし直線と円、あるいは円と円が交わったときの角度はどのように定義すべきかと述べています。そしてその答えとして、２個の円が交わる点Pのそれぞれの接線が作る角度が円と円の交わった場合の角度であると述べています。 従いまして掲載の文章にはどうすることがinversionなのかの説明はございません。また円に関してinversionがあってもなぜ角度を保持するのかの説明もございません。
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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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8.4.3 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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8.3.5 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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微積分の問題です。問題は英語で書かれています。 原文: Show that the angle of incidence equals the angle of reflection for the parabola y^2=4x at the point (0.5, √2). The angle of incidence is measured between the horizontal line through this point and the tangent line at this point. The angle of reflection is measured between the focal line to this point and the tangent line at this point. Note: The focus for this parabola is at (1, 0). 日本語訳: 放物線 y^2=4x 上の点(0.5, √2)での入射角と反射角が同じであることを証明せよ。入射角は、この点を通る水平線とこの点に対する接線との間の角度を指し、反射角は、この点への焦点線とこの点に対する接線との間の角度を指す。注意: この放物線の焦点は(1, 0)である (と訳してみました)。 先生は tan θ=|(M1-M2)/(1+M1-M2)| を使え、とヒントをくれました。 でも解法は教えてくれませんでした。 自分でやったところまで書きます。 まず、ここでの水平線は y=√2 で良いですか? 放物線上の点(0.5, √2)から焦点(1, 0)への傾きは y(1-0.5)=x(0-√2) 0.5y=-√2x y=-2√2x Y軸との交点 b は傾きに焦点の位置を代入して 0=-2√2*1+b b=2√2 よってy=-2√2x+2√2x 放物線上の点(0.5, √2)の接線は グラフを見ながら勘で y=√2x+√2/2 としてみると なんとぴったりでした。 しかし、理由が分かっていません。 …分かるのはここまでです。 これから先はどうすればよいのでしょうか? どなたか教えてください。よろしくお願いします。
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Can a function in polar form have a tangent at a pole? Obviously, the function itself can pass through the pole at (0,0) But what has to happen to the derivative at this point? I.Consider the function r = cos(2θ) on [0,2π) a) Find the general equation that gives the slope of the tangent line. b) Find both r and dr/dθ at 4π/4 c) Find the tangent line(s) to the function at the pole d) For what values of θ does r = cos(2θ) have a tangent line at the pole?
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POINT LATTICES AND THE UNIT CELL Let’s consider the three-dimensional arrangement of points in Fig.15.This arrangement is called a point lattice. If we take any point in the point lattice it has exactly the same number and arrangement of neighbors(i.e.,identical surroundings) as any other point in the lattice. This condition should be fairly obvious considering our description of long-range order in Sec. 2.1 We can also see from Fig. 15 that it is possible to divide the point lattice into much smaller untils such that when these units are stacked in three dimensions they reproduce the point lattice. This small repeating unit is known as the unit cell of the lattice and is shown in Fig.16 A unit cell may be described by the interrelationship between the lengths(a,b,c) of its sides and the interaxial angles (α,β,γ)between them. (α is the angle between the b and c, axes,β is the angle between the a and c axes, and γ is the angle between the a and b axes.)The actual values of a,b,and c, and α,β and γ are not important, but their interrelation is. The lengths are measured from one corner of the cell, which is taken as the origin. These lengths and angles are called the lattice parameters of the unit cell, or sometimes the lattice constants of the cell. But the latter term is not really appropriate because they are not necessarily constants; for example, they can vary with changes in temperature and pressure and with alloying. [Note: We use a,b and c to indicate the axes of the unit cell; a,b and c for the lattice parameters, and a,b and c for the vectors lying along the unit-cell axes.]
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POINT LATTICES AND THE UNIT CELL Let’s consider the three-dimensional arrangement of points in Fig.15.This arrangement is called a point lattice. If we take any point in the point lattice it has exactly the same number and arrangement of neighbors(i.e.,identical surroundings) as any other point in the lattice. This condition should be fairly obvious considering our description of long-range order in Sec. 2.1 We can also see from Fig. 15 that it is possible to divide the point lattice into much smaller untils such that when these units are stacked in three dimensions they reproduce the point lattice. This small repeating unit is known as the unit cell of the lattice and is shown in Fig.16 A unit cell may be described by the interrelationship between the lengths(a,b,c) of its sides and the interaxial angles (α,β,γ)between them. (α is the angle between the b and c, axes,β is the angle between the a and c axes, and γ is the angle between the a and b axes.)The actual values of a,b,and c, and α,β and γ are not important, but their interrelation is. The lengths are measured from one corner of the cell, which is taken as the origin. These lengths and angles are called the lattice parameters of the unit cell, or sometimes the lattice constants of the cell. But the latter term is not really appropriate because they are not necessarily constants; for example, they can vary with changes in temperature and pressure and with alloying. [Note: We use a,b and c to indicate the axes of the unit cell; a,b and c for the lattice parameters, and a,b and c for the vectors lying along the unit-cell axes.]
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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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