A centrolinead guides perspective lines toward a vanishing point that can lie outside the drawing board. This version has two adjustable guide arms and a straight drawing edge meeting at a common joint. Once set, the arms remain locked together while their edges slide against two fixed board pins.
Centrolinead Interactive Calculator
Choose the vanishing-point distance and the two pin offsets. Watch a rigid instrument slide against the pins, and compare the actual arm settings with the bearings from the vanishing point. The separate deviation input calculates a line offset over the chosen distance.
Equation Used
- Rigid, locked arms with ideal point contact at each pin.
- The active drawing edge passes through the common arm joint.
- The joint follows the right-hand arc of the circle through both pins and V.
- Arm lengths are fixed throughout motion and long enough to retain both contacts.
- The deviation comparison is not a model of misadjusted arms, play or instrument accuracy.
Ideal point-contact geometry of a three-arm drafting instrument. Arms remain locked together and slide against the two fixed pins. Finite pin radius, contact-edge thickness, adjustment hardware and drawing-board limits are simplified. The angular-deviation comparison is separate from the correctly set instrument.
Two fixed pins guide a rigid instrument
Stanley’s 1868 treatise illustrates a straightedge with two adjustable arms meeting at a common axis. Their settings are locked before use; the arms stay against fixed drawing-board studs as the instrument moves. One working edge of the rule passes through that axis.
The animation uses ideal point contacts. The gold pins stay fixed while the tan arm edges slide past them. The arms do not change length or hinge on the pins. The blue drawing edge and its faint extensions aim at V.
The displayed motion covers a limited arc with both contacts maintained. Its dimensions illustrate the geometry; they do not specify the size or adjustment plate of a particular instrument.
Perspective lines toward a remote point
The tool helps draw lines aimed at a vanishing point beyond the practical reach of a straightedge. The faint fan of lines shows that purpose directly. Sliding the instrument selects a different line in the same fan.
The equal-offset arrangement is a useful symmetric case. This calculator also permits unequal pin offsets as an ideal geometric generalization. It does not predict the usable travel of a particular historical instrument.
Bearings, arm settings and the joint path
Place the pins at A = (0,d1) and B = (0,−d2), with V = (−L,0). Their bearings from V are α1 = atan(d1/L) above the horizon and α2 = atan(d2/L) below it.
Those bearings are not the obtuse angles between the forward drawing direction and the guide arms. In this construction the upper arm setting is 90° + α2, and the lower setting is 90° + α1. The included angle between the guide arms is 180° − α1 − α2.
The joint J moves on a circle through A, B and V. Its center is C = ((d1d2−L²)/(2L), (d1−d2)/2), with radius R = √(Cx² + ((d1+d2)/2)²). The inscribed-angle relationships keep both arm settings constant. The rule points along J−V, so its working edge always passes through V when extended.
The separate comparison L tan δ is the transverse offset of a line deviating by δ after advancing a distance L along the reference direction. It is not a prediction of the vanishing-point shift caused by changing one guide-arm setting.
A 36-inch vanishing-point distance
For L = 36 in, d1 = 12 in and d2 = 18 in, the pin bearings are 18.435° and 26.565°. The actual guide-arm settings are 116.565° for the upper arm and 108.435° for the lower arm, measured from the forward drawing direction.
The joint is 6 in to the right of the pin line when it lies on the horizon. As it slides above and below that position, the contact points move along the rigid arms while the drawing edge remains directed toward V.
A separate 0.5° line deviation produces about 0.314 in offset over 36 in. Setting that deviation to zero changes the comparison to zero; it does not change the correctly adjusted instrument geometry.
What the drawing simplifies
Real pins have a radius, and real arm edges have thickness. The ideal model treats each pin as a point lying on a contact edge. The rendered pin and joint sizes are illustrative rather than dimensions for making an instrument.
Clearance, wear, finite board size and imperfect settings limit a real tool. No manufacturing tolerance or drafting-accuracy guarantee follows from this calculator.
Calculator questions
Are the arms pivoted to the board pins?
No. Their edges slide against fixed pins; the common instrument joint moves.
Why are the actual arm settings greater than 90°?
They are measured from the forward drawing direction away from V toward each pin-contact arm. The smaller atan values are bearings from V to the pins.
Why does the deviation slider not bend the instrument?
It is a separate line-offset comparison. The animated instrument remains correctly set for the selected vanishing point and pin positions.
Does the drawing show a measured historical instrument?
No. Its arrangement follows the cited treatise, while the dimensions and unequal-offset geometry are an ideal interactive construction.
References & Further Reading
- W. F. Stanley, A Descriptive Treatise on Mathematical Drawing Instruments, third edition (1868), printed pages 145–149: centrolinead construction, common axis and contact with fixed studs.
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