Focal Distance: The distance of a point \((x_1, y_1)\) on the parabola, from the focus, is the focal distance.The focal chord cuts the parabola at two distinct points. Focal Chord: The focal chord of a parabola is the chord passing through the focus of the parabola.The directrix is perpendicular to the axis of the parabola. Directrix: The line drawn parallel to the y-axis and passing through the point (-a, 0) is the directrix of the parabola.Focus: The point (a, 0) is the focus of the parabola.Some of the important terms below are helpful to understand the features and parts of a parabola y 2 = 4ax. The standard equation of a regular parabola is y 2 = 4ax. The general equation of a parabola is: y = a(x-h) 2 + k or x = a(y-k) 2 +h, where (h,k) denotes the vertex. Parabola is an important curve of the conic sections of the coordinate geometry. "A locus of any point which is equidistant from a given point ( focus) and a given line ( directrix) is called a parabola." Thus, a parabola is mathematically defined as follows: Also, an important point to note is that the fixed point does not lie on the fixed line. The fixed point is called the "focus" of the parabola, and the fixed line is called the "directrix" of the parabola. parabola refers to an equation of a curve, such that each point on the curve is equidistant from a fixed point, and a fixed line. Wagner, K.: Bemerkungen zum Vierfarbenproblem. In: 1st ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. Schnyder, W.: Embedding planar graphs on the grid. Kobourov, S.G.: Spring embedders and force directed graph drawing algorithms. Kleist, L., Klemz, B., Lubiw, A., Schlipf, L., Staals, F., Strash, D.: Convexity-increasing morphs of planar graphs. Hopcroft, J.E., Kahn, P.J.: A paradigm for robust geometric algorithms. įloater, M.S., Gotsman, C.: How to morph tilings injectively. R15–R15 (2004)įloater, M.S.: Parametric tilings and scattered data approximation. The Electronic Journal of Combinatorics, pp. Acta Scientiarum Mathematicarum 11(2), 229–233 (1948)įelsner, S.: Lattice structures from planar graphs. įáry, I.: On straight-line representation of planar graphs. Įrickson, J., Lin, P.: Planar and toroidal morphs made easier. Prentice Hall, Hoboken (1999)Įades, P., Garvan, P.: Drawing stressed planar graphs in three dimensions. ĭi Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. In: 12th Symposium on Computational Geometry (SoCG), pp. CRC Press (2013)Ĭhrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Handbook of Graph Drawing and Visualization, pp. Algorithmica 47(4), 399–420 (2007)Ĭhimani, M., Gutwenger, C., Jünger, M., Klau, G., Klein, K., Mutzel, P.: The open graph drawing framework (OGDF). Keywordsīonichon, N., Felsner, S., Mosbah, M.: Convex drawings of 3-connected plane graphs. A third approach chooses the weight of each edge according to its depth in a spanning tree rooted at the outer vertices, such as a Schnyder wood or BFS tree, in order to pull vertices closer to the boundary. We further explore a “kaleidoscope” paradigm for this xy-morph approach, where we rotate the coordinate axes so as to find the best spreads and morphs. A second approach morphs x- and y-spread drawings to produce a more aesthetically pleasing and uncluttered drawing. One approach constructs weights (in linear time) that uniformly spread all vertices in a chosen direction, such as parallel to the x- or y-axis. We present a number of approaches for choosing better weights. A major drawback of the unweighted Tutte embedding is that it often results in drawings with exponential area. Stress-graph embeddings are weighted versions of Tutte embeddings, where solving a linear system places vertices at a minimum-energy configuration for a system of springs. We study methods to manipulate weights in stress-graph embeddings to improve convex straight-line planar drawings of 3-connected planar graphs.
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