Plane (geometry)

“ While it is true that to a boy or girl a solid is more concrete than a plane, it is not true that a geometric solid is more concrete than a geometric plane. Just as the world developed its solid geometry, as a science, long after it had developed its plane geometry, so the human mind grasps the ideas of plane figures earlier than those of the geometric solid. ”

“ The sphere is less frequently looked upon in geometry as a surface, and in popular usage it is always taken as a solid.
Analogous to the postulate that a circle may be described with any given point as a center and any given line as a radius, is the postulate for constructing a sphere with any given center and any given radius. This postulate is not so essential, however, as the one about the circle, because we are not so concerned with constructions here as we are in plane geometry. ”

“ Definition Geometry deals with the properties, relations, and measurement of lines, surfaces, solids, and angles. Plane geometry deals with plane figures, and solid geometry deals with three–dimensional figures.
A point, considered mathematically, is a place having position but no extent. It has no length, breadth, or thickness. A point in motion produces a line, which has length, but neither breadth nor thickness. A straight or right line is the shortest distance between two points in space. ”

“ In the geometry of two dimensions we can thus have either a plane or a sphere or a pseudosphere which are characterized by the property that a surface may be moved about in all directions without any change either in the lengths of its lines or in the magnitudes of its angles. The axiom which assumes that there is only one geodesic connecting two points marks off the plane and the pseudosphere from the sphere. The axiom that only one parallel can be drawn through a given point to a given line marks off the plane from the pseudosphere. ”

“ But a plane is only one among an infinite number of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead of the plane, and obtain “spherical” in the place of “plane” geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all propositions up to the 28th will be true for these figures. ”

“ PLANE GEOMETRY.—Candidates will be required to give accurate definitions of the terms used in plane geometry, to demonstrate any proposition of plane geometry as given in the ordinary text-books and to solve simple geometrical problems either by a construction or by an application of algebra. ”

“ In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is AB and the point P is taken on AB produced, then the square on any tangent from P is equal to PB × PA. The line PBA is sometimes called the radical axis.
Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. ”

“ As distinct particular cases of the first figure we have the plane curve and the cone, and as a particular case of the second figure the ruled surface, regulus or singly infinite system of lines; we have, besides, the congruence or doubly infinite system of lines and the complex or triply infinite system of lines. And thus crowds of theories arise which have hardly any analogues in plane geometry ”

“ But any plane α cuts each of the planes π1, π2 in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. ”

“ Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. ”

“ It will be noted that Prop. 15 at once allows of the solution of the problem: “Through a given point to draw a plane parallel to a given plane.” And it is also easily proved that this problem allows always of one, and only of one, solution.
§ 78. We come now to planes which are perpendicular to one another. Two theorems relate to them.
Prop. 18. If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane. ”

“ As the plane α is perpendicular to the vertical plane, the trace α′′ contains the projection of every figure in it; the points E2, F2, G2, H2 where this trace cuts the elevations of the edges will therefore be the elevations of the points where the edges cut α. From these we find the plans E1, F1, G1, H1, and by joining them the plan of the section. If from E1, F1 lines be drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane α cuts it; hence also the line EF. Similarly on the other faces. ”

“ Even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane.
§ 19. Some of the simpler problems connected with the representation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface.
To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. ”

“ The tangents drawn from a point P to a quadric surface form a cone of the second order, for the polar plane of P cuts it in a conic.
6. If the pole describes a line a, its polar plane will turn about another line a′, as follows from 2. These lines a and a′ are said to be conjugate with regard to the surface.
§ 100. The pole of the line at infinity is called the centre of the surface. If it lies at the infinity, the plane at infinity is a tangent plane, and the surface is called a paraboloid.
The polar plane to any point at infinity passes through the centre, and is called a diametrical plane. ”

“ Another point on which the present theory throws light on tho ordinary geometry must be here alluded to. We have seen that the two tangents from to the fundamental conic remain unchanged during the rotation of the plane round 0. It certainly does seem paradoxical to assert that a plane, and all it contains are rotated around a point, and that nevertheless this operation does not alter the position of a certain pair of lines in the plane which pass through the point. But have we not precisely the same difficulty in ordinary geometry ? ”

“ Again, a plane does not form a boundary among lines through a point: we can pass from any one such line to any other without passing through the plane. But a great circle does divide the surface of a sphere. So, in the polar form, a complete straight line does not divide a plane, and a plane does not divide space, and does not, like a Euclidean plane, have two sides.18 But, in the antipodal form, a plane is, in these respects, like a Euclidean plane. ”

“ Just as we arrived at the infinitely large sphere by letting a finite sphere grow, so must we consider any finite sphere as having grown from a sphere with infinitely small extension; that is, from a point. To travel from the point to the infinitely distant plane in the sense of projective geometry, therefore, means that we have first to identify ourselves with the point and 'become' the plane by a process of uniform expansion in all directions. ”

“ A plane may be represented in multiple algebra as in synthetic geometry by a single letter; in the ordinary algebra it is sometimes represented by three coordinates, for which it is most convenient to take the reciprocals of the segments cut off by the plane on three axes. But the modern analyst has a more ingenious method of representing the plane. ”

“ What are the differences between the geometry on a sphere and the geometry on a plane? If you make a triangle on the earth, whose sides are great circles, you will not find that the angles of the triangle add up to two right angles: they will add up to rather more. The amount by which they exceed two right angles is proportional to the size of the triangle. ”

“ Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it. So our space may also curve through some higher space without our being conscious of it. If our planes have really some curvature, then the sum of the angles of our triangles has a slight excess over 180°. ”

“ It can be proved that any two planes, with which any two of the ideal points are both coherent, will serve as the guiding planes used in the definition. The ideal planes are defined as in projective geometry, and all the other definitions (for segments, order, &c.) of projective geometry are applied to the ideal elements. If an ideal plane contains some proper ideal points, it is called proper, otherwise it is improper. ”

“ The reader will at once perceive the analogy to a well-known phenomenon in ordinary geometry. The equation in rectangular coordinates a; 2 + 7/ 2 =0 denotes either a circle of which the radius is zero or the pair of lines in the latter case we are obliged to admit that the distance of any point on either of these lines from their intersection is equal to zero. We have now to consider the displacement of a rigid figure, and we shall for the present speak only of a plane movement. ”

“ For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value , which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value i.e. a smaller value than , the difference being the more considerable, the greater is the radius of the circle in comparison with the radius R of the “world-sphere.” ”

“ Representation of Points.—We have thus the following method of representing in a single plane the position of points in space:—we take in the plane a line xy as the axis, and then any pair of points A1, A2 in the plane on a line perpendicular to the axis represent a point A in space. If the line A1A2 cuts the axis at A0, and if at A1 a perpendicular be erected to the plane, then the point A will be in it at a height A1A = A0A2 above the plane. ”

“ Thus the locus of a point so moving in a plane as constantly to be at a given distance from a fixed point in the plane is a circle. In analytic geometry we usually speak of the locus of a point, thinking of the point as being anywhere on the locus. Some teachers of elementary geometry, however, prefer to speak of the locus of points, or the locus of all points, thus tending to make the language of elementary geometry differ from that of analytic geometry. ”