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Diff — Parabola

Revision #1454 → #1982 · back to history

addedArchimedes' quadrature of the parabola925f0783c6e6
modifiedArc length of a parabola1ad486276cd6
FieldFrom #1454To #1982
anchors[{"section":"Arc length","snippet":"the lengths of arcs of the parabola that terminate at X can be calculated from f and p as follows"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}h&={\\frac {p}{2}},\\\\q&={\\sqrt {f^{2}+h^{2}}},\\\\s&={\\frac {hq}{f}}+f\\ln {\\frac {h+q}{f}}.\\end{aligned}}}"}][{"section":"Arc length","snippet":"the lengths of arcs of the parabola that terminate at X can be calculated from f and p as follows"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}h&={\\frac {p}{2}},\\\\q&={\\sqrt {f^{2}+h^{2}}},\\\\s&={\\frac {hq}{f}}+f\\ln {\\frac {h+q}{f}}.\\end{aligned}}}"}]
modifiedQuadratic Bézier curve0a431c517b71
FieldFrom #1454To #1982
anchors[{"section":"As quadratic Bézier curve","snippet":"A quadratic Bézier curve is a curve"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\vec {c}}(t)&=\\sum _{i=0}^{2}{\\binom {2}{i}}t^{i}(1-t)^{2-i}{\\vec {p}}_{i}\\\\[1ex]&=(1-t)^{2}{\\vec {p}}_{0}+2t(1-t){\\vec {p}}_{1}+t^{2}{\\vec {p}}_{2}\\\\[2ex]&=\\left({\\vec {p}}_{0}-2{\\vec {p}}_{1}+{\\vec {p}}_{2}\\right)t^{2}+\\left(-2{\\vec {p}}_{0}+2{\\vec {p}}_{1}\\right)t+{\\vec {p}}_{0},\\quad t\\in [0,1].\\end{aligned}}}"}][{"section":"As quadratic Bézier curve","snippet":"A quadratic Bézier curve is a curve"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\vec {c}}(t)&=\\sum _{i=0}^{2}{\\binom {2}{i}}t^{i}(1-t)^{2-i}{\\vec {p}}_{i}\\\\[1ex]&=(1-t)^{2}{\\vec {p}}_{0}+2t(1-t){\\vec {p}}_{1}+t^{2}{\\vec {p}}_{2}\\\\[2ex]&=\\left({\\vec {p}}_{0}-2{\\vec {p}}_{1}+{\\vec {p}}_{2}\\right)t^{2}+\\left(-2{\\vec {p}}_{0}+2{\\vec {p}}_{1}\\right)t+{\\vec {p}}_{0},\\quad t\\in [0,1].\\end{aligned}}}"}]
addedTriple-angle cubic for the trisectrix construction0a0914f989e5