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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1152.0. "Tractrix" by DEC25::ROBERTS (Reason, Purpose, Self-esteem) Tue Nov 21 1989 16:03

    I used to be able to do this with my eyes tied behind my back. Now,
    it's more of a struggle.
    
    Imagine a string, one unit in length, initially placed on the X-Y
    coordinate system such that one end, A, is located at (0,0) and the
    other end, B, is located at (0,1). What are the coordinates of B after
    the string has been pulled by end A along the X-axis to (t,0)?
    
    /Dwayne

T.R	Title	User	Personal Name	Date	Lines
1152.1		KOBAL::GILBERT	Ownership Obligates	`Tue Nov 21 1989 16:58`	13
	I assume that what you really want has all the 'drag' at B, not along the length of the string. That being so, the differential equation for the curve is: dy - y -- = ---------- dx sqrt(1-y�) There's gotta be some better way. FWIW: A hanging chain forms a catenary. If you cut the catenary at its lowest point, the cut end will trace out the above tractrix (from "Mathematical Snapshots").
1152.2	Cat & Mouse	DEC25::ROBERTS	Reason, Purpose, Self-esteem	`Wed Nov 22 1989 11:20`	17
	RE: 1152.1 "I assume that what you really want has all the 'drag' at B, not along the length of the string." Quite right. Another way to express the problem is: Mickey Mouse leaves his house and takes a walk due east. Sylvester the cat, initially one unit north of Mickey, stalks him, remaining exactly 1 unit distant. At any instant, Sylvester is always moving directly toward Mickey. Where is Sylvester when Mickey is t units away from home? /Dwayne
1152.3	Looked it up	AKQJ10::YARBROUGH	I prefer Pi	`Wed Nov 22 1989 14:58`	10
	The equation of a tractrix symmetric about the y-axis and asymptotic to the x-axis is 2 2 2 2 x = (a arccosh(a/y)-sqrt(a -y )) where a>0 is the height of the curve above the x-axis at x=0. (From Bronshtein & Semendyayev, 'Handbook of Mathematis', p.85). Yeah, it's messy. Lynn
1152.4		DEC25::ROBERTS	Reason, Purpose, Self-esteem	`Wed Nov 22 1989 16:33`	4
	Is anyone able to derive it? /Dwayne
1152.5	Derive by showing it works?	VMSDEV::HALLYB	The Smart Money was on Goliath	`Wed Nov 22 1989 16:34`	1
	Should you be able to differentiate .3 and get .1 ?
1152.6		AITG::DERAMO	Don't stop short of the peak!	`Wed Nov 22 1989 17:02`	6
	>> Should you be able to differentiate .3 and get .1 ? I believe so, with a = 1. If not, one of them is probably wrong. Dan
1152.7		KOBAL::GILBERT	Ownership Obligates	`Wed Nov 22 1989 17:09`	2
	Yes, that fits the equation in .1 nicely. Note that it's most easily done with x treated as a function of y.