Introduction: Parabola Tracing Push CarT
This drawing tool/ instrument is meant for tracing parabolas of variable parameters. Although, in the modern age there are printers available that can do the job of printing any shape (mathematical shapes of course included), the motivation for designing this instrument was the fact that there are still instruments like compass (for drawing circle) that are required for drawing on paper with hand. I imagine that, for instance, this instrument could be used by students and maybe professionals of engineering drawing, design or architecture.
The designing of this instrument has gone through many iterations. It all started maybe more than a decade ago when I was planning to make a parabolic reflector mirror for heating water. At that time I thought that a string dipped in ink and suspended by its two ends and hanging like a garland should form a parabolic shape. So instead of getting to work and marking this shape on paper, I got distracted by the question that "Is it possible to prove that this garland shape would be a parabola and if yes, what would its equation be in terms of the string length, gravitational acceleration etc.?". So after a few months of playing with calculus, mechanics, energy minimisation concepts, I did figure out to my delight, that this hanging thread has the shape of hyperbolic cos, denoted by y = cosh x, and not that of a parabola (y = a x^2). This is how my research began for designing a purely mechanical instrument that could draw a parabola, without the use of printers, and it would be an addition to the family of other hand-operated drawing tools like compass, scale, and conic sections drawing tools.
The design and materials of this prototype have been kept as basic as possible so that the functionality can be demonstrated. If one wants, then some of the parts can be made with better design and materials. Also some of the dimensions provided here may have to be changed depending on the material that you have access to.
The features and basic working of this tool are-
- The tool has an upturned T shape and looks like a crane with four wheels at its corners.
- The tool needs to be dragged like a toy car over the paper and the pencil/ pen traces the parabola.
- The tool is suitable to be used on a table top or even the floor.
- The paper does not have to be pierced anywhere. It just has to be secured on the table or floor with tape or other means.
- One shortcoming is that the instrument can trace only half of the parabola in a single run. To trace the other mirror image half of the parabola, some rearrangements of ribbons are required to be done and the instrument then has to be dragged in the opposite direction.
- The mechanism relies on two sliders constrained to move on two tracks which are the segments of the upturned T.
- The movement of the sliders is controlled by three ribbons which provide the required forces and mathematical constraints.
- As the cart like instrument is moved horizontally (along x- axis) over the paper, the vertical slider (z-axis) exactly follows the horizontal movement of the instrument.
- The slider that moves on the horizontal track (y-axis) has the pen, and the slider on the vertical track constrains the movement of the horizontal slider via a ribbon.
- The mathematics of the working is given in the last section.
I have named the instrument - GAUTAM (!) which is an acronym for- 'Gravity-Assisted' refers to the use of heavy nuts for providing forces to various elements, rather than using motors or springs; 'Upturned-T' refers to the shape of the tool; 'Arch-Maker' refers to the parabola shape that the tool draws.
In the instructions to make the assembly, I have shown photographs of the various assembled parts rather than showing how the components are to be assembled. This is because the photographs are quite self explanatory. The steps will be explained in the written part.
The working is shown in a video in this step.
All the parts listed here have not been separately photographed, but can be seen used in various parts of the instrument.
- Four pieces of F-shaped, anodised aluminium extruded channels (generally used for windows)
- Blocks of plywood - 18 mm thick
- Thin plywood sheet - 5 mm thick
- Castors (generally used for window panels)
- Plastic tube - 6.5 mm OD
- Screws - 4 mm dia, 1/2" to 2" - various lengths; and small nuts and washers.
- Heavy nuts as weights
- Hard Wire GI wire.
- Thin nails, and common nails.
- Narrow Ribbon that is used in craft, or any other non stretchable, yet thin cloth strip.
- Ear bud that acts as a smooth marker when dipped in ink.
- Straws or ball pen body or other hollow cylindrical objects that could be made into pulleys
- Saw for cutting the plywood,
- Hack saw for cutting the aluminium channels
Step 1: Assembly: Main Structure
The photographs show the various parts of the instrument and their names. There are eight (8) main structural parts and three ribbons. These parts are labelled in the first photograph.
- The horizontal track (Along the y-axis)
- The vertical track (Along the z axis)
- The horizontal slider that slides on the horizontal track and holds the pencil/ pen for tracing
- The vertical slider that slides on the vertical track
- The vertex end joint
- Parabola Open End (POE) joint
- The T-junction joint
- The Top joint
- 'Red' Ribbon pulls the horizontal slider along the track. Actually all three ribbons are identical and pink in colour, but to distinguish them these colors have been assigned in the schematic( given in the last photo)
- The 'black' ribbon lowers the vertical slider in sync with the horizontal movement of the instrument on the table.
- The 'light blue' ribbon constrains the horizontal movement of the horizontal slider, depending on the movement of the vertical slider. The mathematics is given in the last section.
The horizontal and vertical tracks are made with anodised, F-shaped aluminium channels. The lengths of the channels are: 62 cm horizontal channels, and 46 cm vertical channels. The dimensions of the cross section of the F channel: Height of the 'F' = 20 mm, width of the 'F' = 15 mm.
The plywood strips on which the castors are mounted- 240 x 30 x 9 (mm)
The width of the plywood blocks for separating the channels within a pair: 56 mm
The height of the triangular plywood pieces= 12 cm, width at the base= 10 cm, width at the top= 2 cm. the position of the holes for inserting screws is not critical.
The height of the horizontal track channel is about 5 cm above the ground.
The dimensions are not sacrosanct, one can redesign their own instrument based on size of materials available. The pulleys can also be made in a better way. Other possibilities are - Using rectangular extruded aluminium channels instead of wooden pieces.
Step 2: The Y- Axis Slider (marker Holder)
The dimensions of the y-slider that holds the marker: 30 x 30 x 10 mm cuboid, with a hole (10 mm dia) drilled off-center so that the wire over which the 'light blue' ribbon passes over is directly above the hole. The hole could have been smaller too. I just made it big so that whatever hollow cylindrical objects were available could be experimented with.
Four thin nails have been hammered near the corners. To these nails two thick GI wires have been stuck with Araldite. The ends of the GI wire are inserted into a plastic tube of 6.5 dia whose photo is shown- it is seen within the channel. To make the horizontal slider movement smooth, I applied machine oil to the inside of the anodised aluminium channel.
Within the 10 mm hole in the cuboid, a cylindrical pen cap fits snugly and within this pen cap another refill tube is fixed by using newspaper cladding. Within this refill tube the earbud slides freely.The cotton is dipped in ink just before drawing.
Step 3: Assembly: the Z-axis Slider
The parts required are-
- plywood sheet of 5 mm thickness
- long screw (1.5"- 2", 4 mm dia ) with three nuts, and small and medium size washers as shown
- 2 large nuts as weights
- The sizes are shown in the photographs.
- Follow the photographs in sequence to understand the procedure. Basically the red and light blue ribbons should be attached to this slider. The slider should be inserted into the track and then the other pieces should be attached to it.
Step 4: Assembly: Incorporating the Three Ribbons
The steps refer to the photographs in the same sequence.
- The schematic where all the ribbons are shown with color coding and their spatial arrangement, without the other structural parts. The black ribbon is anchored to an external object whereas the light blue ribbon is anchored to one end of the instrument itself (on the POE end). The later references to 'red' ribbon, 'light blue' ribbon and 'black' ribbon are to be understood by considering this schematic, since the actual colour of all ribbons is pink, else it can get confusing!
- The 2nd picture is of the 'red' ribbon. The ribbon passes over a pulley installed on the TOP junction, and four nuts are suspended at the open end of the ribbon as weights.
- The red ribbon then passes around the pulley attached at the Vertex end junction (3rd photo),
- and the other end is attached to the horizontal slider by looping it around the metal wire and stapled, as shown in 4th photo.
- The 5th picture is of the 'black' ribbon. One end is attached to the top of the vertical slider and,
- the ribbon passes over two pulleys installed on the TOP junction.
- The ribbon comes down and passes around a pulley installed in the T-junction.
- The other end of the 'black' ribbon is then anchored to a heavy object like a text book by clamping it between the pages and placing a heavy object on top of the book.
- One end of the 'light blue' ribbon can either be wound on a spool or clamped to the wooden strip on which the castors are mounted, on the POE end.
- 10th photo- Another view of the 'light blue' ribbon with the castors visible.
- It should then be passed over the second pulley of the horizontal slider and the other end of the 'light blue' ribbon should be attached to the vertical slider.
- The 'light blue' and the 'black'ribbon are visible in this 12th photo.
Step 5: The Mathematics Behind the Working
A parabola can be constructed in multiple ways. One of the rules for constructing a parabola is - draw a straight line called directrix, and draw a point (Focus) at a distance 2a from the directrix. Then draw all such points that are equidistant from the focus and directrix. Refer to diagram 1. There are instruments that have been made centuries ago, using this property.
Another well known property of a parabola can be derived from the previous property. Refer to the diagram 2.
The instrument's working relies on the constancy of length of the sum of the two segments CA and AF for all points below the arbitrary line PQ. The constraining ribbon (light blue) is arranged so that the length of the ribbon remains constant. There are again instruments that make use of this property. In such instruments one needs to anchor the focus on the paper, but in the instrument GAUTAM, instead of having the focus in the horizontal plane and at a fixed position, it has been shifted to the vertical plane. The vertical slider plays the role of focus. One can consider it as an equivalent focus or virtual focus- whatever name makes sense. This is a key difference.
Let us see how to convince ourselves that the horizontal y-slider actually traces a parabolic path, and what is the role of the vertical slider in ensuring that the horizontal y- slider does move as required.
Refer to the third diagram.-
Since the length of the 'light blue' ribbon is constant, we have-
sqrt (y^2 + z^2) + (C - y) = D, Here C is a length, not the point that was named C in the previous diagram.
=> y + (D-C) = sqrt (y^2 + z^2),
=> y + 2a = sqrt (y^2 + z^2)
=> y^2 + 4ay + 4a^2 = y^2 + z^2, (squaring both sides)
=> 4ay + 4a^2 = z^2
=> y + a = z^2 / (4a) (dividing both sides by 4a)
This is the equation that relates the displacements y and z of the respective sliders. Also due to the construction of the instrument, the z movement is the copy of the x movement of the instrument across the table(refer to the 4th diagram). Hence we can replace z in the above equation by x, and we get -
y + a = x^2 / (4a)
calling y + a = Y, we get-
Y = x^2 / (4a)
This is the familiar equation of a parabola. Y is the conventional distance along y axis, measured from the vertex, and y is a quantity that we started our analysis with, but in the end everything ties up and makes sense(refer to the 5th diagram). We can conclude that the movement of the y-axis slider should be proportional to the square of the distance traveled along x-axis. And since the y-axis slider is moving simultaneously in the x and y directions, due to the instrument moving in the x-direction and the slider moving in the y-direction, the net path traced will obviously be a parabola.
Step 6: How to Use GAUTAM for Tracing Various Shaped Parabolas
Parabolas can be sharp at the vertex end (like a rocket or bullet) or could be blunt at the vertex end (like a bowl or saucer). The sharpness is decided by the parameter 'a' in the equation of the parabola-
y = x^2 / 4a, -(1)
Smaller the value of 'a', sharper will be the parabola.
'a' is actually the distance between the vertex and the focus. Refer to the 1st diagram.
Calculating a few values needed for obtaining the desired parabola size and shape
In case you want to draw a parabola which you want it to be 'w' cm wide and 'h' cm tall, with the vertex at the bottom, then the corresponding 'a' can be found out by rearranging the terms of the above equation. And we get-
a = x^2 / 4y, -(2)
Here x is actually equal to w/2, and y is just equal to 'h'.
Substituting x = w/2, and y = h, in equation (2) we get-
a = w^2 / 16 h, -(3)
The parabola that has been traced (seen in the 2nd photograph) was made by setting a = 11 cm. The measured values of width and height of the traced parabola are- width(h)= 33 x 2 = 66 cm and height(h) = (24.5 cm). When these are substituted in equation (3), we get a value of a = 11.1 cm. Quite close !
Setting the ribbon lengths (or equivalently the position of sliders)
So to start with, first set the length of the 'blue' ribbon to be equal to D = C + 2a. Equivalently this can be set by placing the y-slider towards the vertex end and setting the distance between the y-slider and the T-junction to be 'a' and affixing the loose end of the 'light blue' ribbon to its anchor on the POE side of the instrument. Refer to the 3rd diagram.
If you want the half width of your parabola to be w/2 cm, then you need to ensure that the starting position of the z-slider should be w/2 cm above its lowest position, the T-junction (Refer to the 4th diagram). With the z-slider at this position, gently make the black ribbon taut and anchor its free end between heavy books.
If you require the height of the parabola to be h cm, then the starting position of the y-slider should be (h-a) cm measured from the T-junction (again refer to the 4th diagram).
Once the positions of the sliders are fine and you have wet the tip of the earbud with few drops of ink, the instrument is ready to draw a parabola. Only a gentle push is required as seen in the video. The black ribbon provides the force to pull the instrument towards the external anchor (the pile of books).
The way to understand the working is by considering it as a 5-6 step process-
- We gently push the instrument along the x-axis, towards the anchor of the black ribbon.
- This causes the 'black' ribbon to slack and it allows the vertical z-slider to move down (by a distance equal to the distance that the instrument moved in the x-direction).
- Due to the z-slider moving down, it slackens the 'light blue' ribbon.
- Due to the 'light blue' ribbon losing its tension, the horizontal slider experiences an unbalanced force from one side - that is from the 'red' ribbon alone.
- As a result the weight attached to the free end of the 'red' ribbon lowers, pulling the y-slider towards the vertex end.
- The y-slider moves to an extent till the blue ribbon gets taut again.
- The amount by which the y-slider moves, is decided by the constraint that we discussed in the previous section.
- This process repeats, and one can think of it like the infinitesimal steps that we talk of in calculus.
It kind of amazes me that even though there are so many processes involved, yet the trace of the parabola does not look jagged. It is quite smooth as seen in the 2nd photo.
Do not try to draw the parabola starting from its vertex. Start from the open end of the parabola and move towards the vertex. The instrument is designed in such a way that the two weights should move downward during its operation, then the movement of the y-slider will be smooth.
Now we will see how to reset the sliders -
Once a run has been made and maybe you are not happy with it, then it takes some effort and technique to set up the sliders back to their start positions The weight on the 'red' ribbon and the z-slider are at their bottom positions. A way to reset the setup is-
- Lift the weight hanging on the 'red' ribbon by a few cm.
- Pull down the 'black' ribbon segment that runs down the entire height of the instrument making the vertical slider to move up.
- Then push the entire instrument in the x-direction,away from the 'black' ribbon anchor (books) by a few cm, so that it moves in the direction of its wheels/ castors.
- Repeat this sequence a few times till you are content with the positions.
One could think of ways to make this resetting procedure easier by adding some ribbon securing parts as an improvement to the design.
Participated in the
Made with Math Contest