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2025-06-05 17:51本頁(yè)面
【正文】 continuing through Ptolemy, Copernicus and Galileo, and today with our work on the International Space Station, man continues to learn more and more about the heavens. From here, we look inward to biochemistry and biology. To truly understand biochemistry, scientists study and see the unseen by studying the chemistry of biological processes. This science, along with biophysics, aims to bring a better understanding of how bodies work – from how we turn food into energy to how nerve impulses transmit. analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic. Its most mon application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax+by+c=0 represents a straight line in thexyplane, and the linear equation ax+by+cz+d=0 represents a plane in space, where a, b, c, and dare constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, ., plotting curves and surfaces and determining points of intersection. In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes。 technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the xaxis. If the line is parallel to the xaxis, its slope is zero. Two or more lines with equal slopes are parallel to one another. In general, the slope of the line through the points (x1, y1) and (x2, y2) is given by m= (y2y1) / (x2x1). The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation ax2+bxy+cy2+dx+ey+f=0, where a, b,?…?, f are constants and a, b, and c are not all zero. In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x, y, and zaxes, respectively。 these satisfy the relationship λ2+μ2+ν2= 1. In the same way that the conic sections are studied in two dimensions, the 17 quadric surfaces, ., the ellipsoid, paraboloid, and elliptic paraboloid, are studied in solid analytic geometry in terms of the general equationax2+by2+cz2+dxy+exz+fyz+px+qy+rz+s=0. The methods of analytic geometry have been generalized to four or more dimensions and have been bined with other branches of geometry. Analytic geometry was introduced by Ren233。Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry. circle, closed plane curve consisting of all points at a given distance from some fixed point, called the center. A circle is a conic section cut by a plane perpendicular to the axis of the cone. The term circle is also used to refer to the region enclosed by the curve, more properly called a circular region. The radius of a circle is any line segment connecting the center and a point on the curve。 the term is also used for the length r of this segment, ., the mon distance of all points on the curve from the center. Similarly, the circumference of a circle is either the curve itself or its length of arc. A line segment whose two ends lie on the circumference is a chord。 a chord through the center is the diameter. A secant is a line of indefinite length intersecting the circle at two points, the segment of it within the circle being a chord. A tangent to a circle is a straight line touching the circle at only one point, the point of contact, or tangency, and is always perpendicular to the radius drawn to this point. A circle is inscribed in a polygon if each side of the polygon is tangent to the circle。 a circle is circumscribed about a polygon if all the vertices of the polygon lie on the circumference. The length of the circumference C of a circle is equal to π (see pi) times twice the radius distance r, or C=2πr. The area A bounded by a circle is given by A=πr2. Greek geometry left many unsolved problems about circles, including the problem of squaring the circle, ., constructing a square with an area equal to that of a given circle, using only a straight edge and pass。 it was finally proved impossible in the late 19th cent. (see geometric problems of antiquity). In modern mathematics the circle is the basis for such theories as inversive geometry and certain nonEuclidean geometries. The circle figures significantly in many cultures. In religion and art it frequently symbolizes heaven, eternity, or the universe.
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