įor a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints of the two diagonals. asymptotes, which a curve approaches arbitrarily closely without touching it.i-secant lines, meeting the curve in i points counted without multiplicity, or.In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other.įor more general algebraic curves, lines could also be: a directrix, whose distance from a point helps to establish whether the point is on the conic.exterior lines, which do not meet the conic at any point of the Euclidean plane or.secant lines, which intersect the conic at two points and pass through its interior.tangent lines, which touch the conic at a single point.For instance, with respect to a conic (a circle, ellipse, parabola, or hyperbola), lines can be: However, lines may play special roles with respect to other objects in the geometry and be divided into types according to that relationship. In a sense, all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart from one another. The red line is tangential to the curve at the point marked by a red dot. In the geometries where the concept of a line is a primitive notion, as may be the case in some synthetic geometries, other methods of determining collinearity are needed. However, there are other notions of distance (such as the Manhattan distance) for which this property is not true. In Euclidean geometry, the Euclidean distance d( a, b) between two points a and b may be used to express the collinearity between three points by: The points a, b and c are collinear if and only if d( x, a) = d( c, a) and d( x, b) = d( c, b) implies x = c. By extension, k points in a plane are collinear if and only if any ( k–1) pairs of points have the same pairwise slopes. In particular, for three points in the plane ( n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero.Įquivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal the other slopes). Lines can be referred by two points that lay on it (e.g., A B ↔ The word line may also refer to a line segment in everyday life, which has two points to denote its ends. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. In geometry, a line is an infinitely long object with no width, depth, or curvature.
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