Amsterdam : North-Holland Pub.

Includes bibliographical references and index. Electronic reproduction. Cornelis , Advanced plane geometry.

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Advanced geometry of plane curves and their applications. Mineola, N. Digital Library Federation, December View online Borrow Buy Freely available Show 0 more links None of your libraries hold this item. Found at these bookshops Searching - please wait In kinematics, for example, it will usually be the time, while in purely geometrical problems the length of the curve, measured from a fixed point will be preferably used, especially in considerations of a more general nature.

Further, the abscissa x or the ordinate y and even the radius of curvature may occasionally be the most suitable one.

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## The Advanced Geometry Of Plane Curves And Their Applications

Again, in electrotechnical problems the usual one is the angular frequency along the contours representing impedances as functions of the frequency and we shall, besides, meet many cases, in which still other quantities will play the part of parameter. By changing from one parameter u to another v, being a real function of u , we do not change the character of the curve, the only thing that changes geometrically is the scale along the curve.

It is therefore possible to represent one curve by different equations and we shall in each case choose the one that is most adapted to the problem in hand. In order to familiarize the reader with the above notions we shall by way of introduction treat a number of curves and their analytical representation.

It will strike us that simple functions always give rise to simple curves and vice versa. Its direction is the same as that of the imaginary axis y -axis. The u -scale is uniform, but we may change to non-uniform scales by a transformation of the parameter. Each function. Anticipating later results, we may well state that the simple analytical representation of the circle is the reason why complex calculus is so well-adapted to the treatment of technical problems.

## ISBN 13: 9780486442761

When x is zero one sees that y must be 6, so the graph also passes through 0, 6. It turns out that this equation represents a particular example of a parabola. Once one sees the magic of analytical geometry lots of what is done in algebra for some people makes a lot more sense and certainly is very useful for connecting algebra to geometry via analytical geometry.

Many very interesting curves can be gotten with equations that use trigonometric functions, logarithmic, or exponential functions, in addition to the more familiar algebraic functions. Technical note : The coordinate system used above is often called Cartesian or a rectangular coordinate system. As a matter of historical interest, Descartes did not actually use a system where there were four quadrants, but used only what today would be called the first quadrant, where the values of x and y are both positive.

One could also use what are called polar coordinates to write down equations and get interesting pictures, too. Polar coordinates rely on ideas from trigonometry. Another type of coordinate system would be to use homogeneous coordinates for points and work in what is sometimes called the real projective plane. Thus, in this coordinate system 1, 2, 1 and 2, 4, 2 would be the same point.

For a very brief primer of these ideas look here.

It should be intuitively clear that if we graph two lines that are not sitting on top of each other, the two lines have the same shape. This means that one can either shift translate one line onto the other in which case the lines are called parallel lines or one can locate the point where the two lines intersect and rotate one of the lines about that point until the line coincides with the other line. Given a single point, like the origin, there are infinitely many lines that pass through this point e. However if one picks any two different points, there is only a single line that passes through them.

However, it is especially helpful to see this in the equation:. The constant m is known as the slope of the line. Part of the reason for interest in the concept of slope is that when Newton and Leibniz developed the ideas that became what today is called Calculus, they used the notions of slope of a line and the concept of tangent line to a curve as part of how to conceptualize about derivatives. The derivative helps one understand the important concept of rate of change.

Rates of change come up in physics, chemistry, biology and economics but though Calculus has many applications it has internal questions of interest in its own right. In addition, to understand the "shape" of lines, analytical geometry is also concerned with understanding the "essence" of the properties that make a circle a "circle" or an ellipse a shape different from a "hyperbola. An interesting type of graph which came to be studied in ancient times is known as the hyperbola.

Apollonius wrote a book about the conics over years ago! Hyperbolas are a special kind of interesting class of curves that arise as "conic sections. When this cone is cut by various planes one gets points that lie in the intersection of the cone and the plane.

## Daigle : Local trees in the theory of affine plane curves

When the intersection of the plane and the cone is 2-dimensional one can get a circle, an ellipse, a parabola or the two pieces of a single hyperbola one on the upper part of the cone and one on the lower part. I will postpone until later that one can cut a cone in a very special way to get a single point or a single line, which are often called degenerate conic sections.

I will say more about them below. Figure 7 helps clarify how some of the different kinds of conic sections arise from plane sections of one nape of a cone when the plane is at different angles to the axes and generating lines of the cone. All Euclidean distance circles round of the same radius are congruent, that is, if they are moved without "distortion," any circle of radius r can be superimposed on any other circle of radius r.

However, it may seem mysterious that the same circle has different equations depending on where the coordinate system is placed with regard to the circle. A circle referred to a pair of orthogonal meet at right angles axes at the point 0,0 has the equation below. The center of the circle is at 0, 0 the origin.

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Given such equations one can look for points that satisfy the equation and thus lie on the graph of the curve that the equation defines. A complication is that there are numbers that satisfy equations that are not real numbers and there are equations which have no real number solutions. However, for real numbers when one multiplies such a number by itself e. Furthermore, adding positive numbers can only yield a positive number.

So there are no real numbers x and y which satisfy 3. By the way, it is worth noting that when the right-hand side of 2 is zero, the only solution of the equation shown is 5, -2 which can be thought of as a degenerate circle with radius 0. One can now try to study, by way of example, a particular curve, say the hyperbola.

Just as circles can differ because they have different radii, hyperbolas can be "congruent" to each other if one moves one curve without "distorting" it, one can superimpose it on another curve in a different location but they can also differ in shape. In Figure 8 the different colored curves are not congruent, though they all "look" like hyperbolas. Different shaped curves in the same family will have different equations.

If one refers the curves to the degree angle line shown there, and a line is drawn through 0, 0 perpendicular to that line, one will get a more familiar equation for the hyperbolas involved:. For each point on the hyperbola the tangent line locally the line meets the curve in exactly one point to the hyperbolas cuts off a triangle for which the x -intercept and the y -intercept when multiplied, yields a constant value. This fact holds for both the case where the coordinate axes meet at right angles or where they meet at other angles.

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A nifty application of that fact is to examine those points in the interior of a triangle where there is a line through the point which bisects the area of the triangle. It turns out that pieces of three hyperbolas are involved in understanding the nature of this set of area bisectors. A similar fact about parabolas enables one to look at the set of points in a triangle through which a perimeter bisector of the triangle passes. In addition to ellipses and hyperbolas, one can obtain circles, parabolas, and intersecting lines by cutting the napes of a cone.

The intersection of a cone and a plane which results in two intersecting lines is considered to be a "degenerate" conic because the geometric object is the product of two 1-dimensional geometric objects rather than a 2-dimensional object per se. And, if one considers a cylinder as a "degenerate" cone, one can also get two coincident lines see below. The complex world of conic sections gets captured by the equation below, where the values of the letters from a to g are real numbers where not all of a, b , and c are zero.

Though this equation gives the general form of a conic section or a degenerate conic section, one rarely sees conics given in this form because mathematical tools can be used to simplify the equation of the conics that one is most interested in--circles, ellipses, hyperbolas, and parabolas. However, one can see an interplay between geometry and algebra, observing that multiplying the equations of two lines together. The two lines one "multiplies" to get a quadratic equation as in 7 can be identical lines or different lines, and the different lines can intersect or they can be parallel.