File:Gunners’ equations.svg - Revision history

Rational Point: update code example

2025-02-04T00:26:28Z

update code example

← Older revision		Revision as of 00:26, 4 February 2025
Line 25:		Line 25:

	svg(); plot(main=c("“gunners’ equations” solved to fit a parabolic trajectory",		svg(); plot(main=c("“gunners’ equations” solved to fit a parabolic trajectory",
	"to a quartic curve given one of two points of tangency"),		"over a quartic curve given one of two points of tangency"),
	x=c(rev(x_data),-x_data,-rev(x_data),x_data),		x=c(rev(x_data),-x_data,-rev(x_data),x_data), xlab="x",
	y=c(Y(rev(x_data)),Y(-x_data),-Y(-rev(x_data)),-Y(x_data)),		y=c(Y(rev(x_data)),Y(-x_data),-Y(-rev(x_data)),-Y(x_data)), ylab="y",
	~~xlab=c(paste("x1 = ", x1, " x = ", x),~~
	~~"y^2 = -10/51x^4 - 40/357x^3 + 8045/2499x^2 + 220/119x + 220/833"),~~
	ylab=~~c(paste("y1 = ", y1, " y = ", y),~~ "y ~~= ax^2 + bx + c~~"),
	type="l", lwd=3, col="brown", asp=1, ylim=c(-4.25,5.75))		type="l", lwd=3, col="brown", asp=1, ylim=c(-4.25,5.75))
	abline(0,0)		abline(0,0)
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	abline(v=(x + x1)/2, col="purple")		abline(v=(x + x1)/2, col="purple")
	abline(y1 - x1*(y - y1)/(x - x1), (y - y1)/(x - x1), col="purple")		abline(y1 - x1*(y - y1)/(x - x1), (y - y1)/(x - x1), col="purple")
	~~text(-1.75,c(-3.5,-3.875,-4.25),labels=c(paste("a = ", a),paste("b = ", b), paste("c = ", c)))~~

	x_vals <- seq(-7,7,0.001)		x_vals <- seq(-7,7,0.001)
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	abline(y1 - x1*y1_, y1_, col="red")		abline(y1 - x1*y1_, y1_, col="red")
	abline(y - x*y_, y_, col="red")		abline(y - x*y_, y_, col="red")

			text(0,c(5.9, 5.5, 5.1, 4.7),labels=c("y = ax^2 + bx + c",
			paste("a = ", a),paste("b = ", b), paste("c = ", c)), col="green")
			text(0,-4.3,labels="y^2 = -10/51x^4 - 40/357x^3 + 8045/2499x^2 + 220/119x + 220/833", col="brown")
			text(x1,c(y1-0.2,y1-0.5),pos=1,labels=c(paste("x1 = ", x1),paste("y1 = ", y1)))
			text(x-0.9,c(y-0.3,y-0.6),pos=1,labels=c(paste("x = ", x),paste("y = ", y)))

	points(c(x1,x),c(y1,y))		points(c(x1,x),c(y1,y))

Rational Point: Rational Point uploaded a new version of File:Gunners’ equations.svg

2025-02-04T00:25:41Z

Rational Point uploaded a new version of File:Gunners’ equations.svg

← Older revision	Revision as of 00:25, 4 February 2025
(No difference)

Rational Point: text of coefficients solved for on chart

2025-02-03T23:37:16Z

text of coefficients solved for on chart

← Older revision		Revision as of 23:37, 3 February 2025
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	"y^2 = -10/51x^4 - 40/357x^3 + 8045/2499x^2 + 220/119x + 220/833"),		"y^2 = -10/51x^4 - 40/357x^3 + 8045/2499x^2 + 220/119x + 220/833"),
	ylab=c(paste("y1 = ", y1, " y = ", y), "y = ax^2 + bx + c"),		ylab=c(paste("y1 = ", y1, " y = ", y), "y = ax^2 + bx + c"),
	type="l", lwd=3, col="brown", asp=1, ylim=c(-4,6))		type="l", lwd=3, col="brown", asp=1, ylim=c(-4.25,5.75))
	abline(0,0)		abline(0,0)
	abline(v=c(x1, x, x1 + (x-x1)/4, x1 + (x-x1)/3, x1 + 2(x-x1)/3, x1 + 3(x-x1)/4), col="yellow")		abline(v=c(x1, x, x1 + (x-x1)/4, x1 + (x-x1)/3, x1 + 2(x-x1)/3, x1 + 3(x-x1)/4), col="yellow")
Line 42:		Line 42:
	abline(v=(x + x1)/2, col="purple")		abline(v=(x + x1)/2, col="purple")
	abline(y1 - x1*(y - y1)/(x - x1), (y - y1)/(x - x1), col="purple")		abline(y1 - x1*(y - y1)/(x - x1), (y - y1)/(x - x1), col="purple")
			text(-1.75,c(-3.5,-3.875,-4.25),labels=c(paste("a = ", a),paste("b = ", b), paste("c = ", c)))

	x_vals <- seq(-7,7,0.001)		x_vals <- seq(-7,7,0.001)
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	c(ax1^2 + bx1 + c - y1, 2ax1 + b - y1_, ax^2 + bx + c - y, 2ax + b - y_)		c(ax1^2 + bx1 + c - y1, 2ax1 + b - y1_, ax^2 + bx + c - y, 2ax + b - y_)
	c(a3a^3 + a2a^2 + a1*a + a0, a, b, c, x)		c(a3a^3 + a2a^2 + a1*a + a0, a, b, c, x)

	</syntaxhighlight>		</syntaxhighlight>

Rational Point: Rational Point uploaded a new version of File:Gunners’ equations.svg

2025-02-03T23:34:56Z

Rational Point uploaded a new version of File:Gunners’ equations.svg

← Older revision	Revision as of 23:34, 3 February 2025
(No difference)

Rational Point: R program to plot chart

2025-02-03T20:37:00Z

R program to plot chart

← Older revision	Revision as of 20:37, 3 February 2025
Line 1:	Line 1:
		<syntaxhighlight lang="R" line="1">
		#! /usr/bin/R -f
		q4 <- -10/51; q3 <- -40/357; q2 <- 8045/2499; q1 <- 220/119; q0 <- 220/833
		Q <- expression(q4x^4 + q3x^3 + q2x^2 + q1x + q0)
		Y <- function(x) {eval(parse(text=paste("sqrt(",Q,")")),list(x=x))}
		Y_ <- function(x) {eval(D(parse(text=paste("sqrt(",Q,")")),"x"),list(x=x))}

		x1 <- -pi; y1 <- Y(x1); y1_ <- Y_(x1)

		a3 <- -8*y1
		a2 <- 4y1_^2 + 12q3x1 + 24q4x1^2 - 4y1_^2 + 4*q2
		a1 <- -4q3y1_ - 16q4x1y1_ + 8q4*y1
		a0 <- q3^2 - 4q4q3x1 - 8q4^2x1^2 + 4q4y1_^2 - 4q4*q2

		a_roots <- polyroot(c(a0,a1,a2,a3))

		a <- a_roots[1]
		b <- y1_ - 2ax1
		c <- y1 + ax1^2 - x1y1_
		x <- -(2ab - q3 +2a^2x1 - 2q4x1) / (2*(a^2 - q4))
		y <- Y(x)
		y_ <- Y_(x)

		x_data <- seq(0,sqrt(16.5),0.001)

		svg(); plot(main=c("“gunners’ equations” solved to fit a parabolic trajectory",
		"to a quartic curve given one of two points of tangency"),
		x=c(rev(x_data),-x_data,-rev(x_data),x_data),
		y=c(Y(rev(x_data)),Y(-x_data),-Y(-rev(x_data)),-Y(x_data)),
		xlab=c(paste("x1 = ", x1, " x = ", x),
		"y^2 = -10/51x^4 - 40/357x^3 + 8045/2499x^2 + 220/119x + 220/833"),
		ylab=c(paste("y1 = ", y1, " y = ", y), "y = ax^2 + bx + c"),
		type="l", lwd=3, col="brown", asp=1, ylim=c(-4,6))
		abline(0,0)
		abline(v=c(x1, x, x1 + (x-x1)/4, x1 + (x-x1)/3, x1 + 2(x-x1)/3, x1 + 3(x-x1)/4), col="yellow")
		abline(y1 - x1(y - y1)/(x - x1) - (x - x1)^2a/4, (y - y1)/(x - x1), col="yellow")
		abline(y1 - x1(y - y1)/(x - x1) - (x - x1)^2a/3, (y - y1)/(x - x1), col="yellow")
		abline(y1 - x1(y - y1)/(x - x1) - (x - x1)^2a/2, (y - y1)/(x - x1), col="yellow")
		abline(y1 - x1(y - y1)/(x - x1) - (x - x1)^2a, (y - y1)/(x - x1), col="yellow")
		abline(y1 - x1*(y1_+(y - y1)/(x - x1))/2, (y1_+(y - y1)/(x - x1))/2, col="orange")
		abline(y - x*(y_+(y - y1)/(x - x1))/2, (y_+(y - y1)/(x - x1))/2, col="orange")
		abline(v=(x + x1)/2, col="purple")
		abline(y1 - x1*(y - y1)/(x - x1), (y - y1)/(x - x1), col="purple")


		x_vals <- seq(-7,7,0.001)
		lines(x_vals, ax_vals^2 + bx_vals +c, lwd=2, col="green")
		abline(y1 - x1*y1_, y1_, col="red")
		abline(y - x*y_, y_, col="red")

		points(c(x1,x),c(y1,y))
		points(c((x + x1)/2, (x + x1)/2, (x + x1)/2),
		c((y + y1)/2, (y + y1)/2 - (x - x1)^2a/4 , (y + y1)/2 - (x - x1)^2a/2))

		c(ax1^2 + bx1 + c - y1, 2ax1 + b - y1_, ax^2 + bx + c - y, 2ax + b - y_)
		c(a3a^3 + a2a^2 + a1*a + a0, a, b, c, x)
		</syntaxhighlight>

		<syntaxhighlight lang="R">
		> c(ax1^2 + bx1 + c - y1, 2ax1 + b - y1_, ax^2 + bx + c - y, 2ax + b - y_)
		[1] 0.000000e+00-4.701977e-38i 0.000000e+00+0.000000e+00i
		[3] 0.000000e+00-6.009324e-22i 4.440892e-16-2.220166e-22i
		> c(a3a^3 + a2a^2 + a1*a + a0, a, b, c, x)
		[1] 0.00000000+1.053066e-22i -0.04593346-1.654361e-23i
		[3] 0.11407143-1.039466e-22i 4.06801052-1.632789e-22i
		[5] 2.88535800-1.260440e-23i
		>
		</syntaxhighlight>

Rational Point at 20:34, 3 February 2025

2025-02-03T20:34:55Z

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