Sine graph
Photo: The first half of the sine graph has been drawn, the tool is rotated 180 degrees so the second half can be created.
Draw an X axis with the ruler. Mark the X=0 and X=180 degrees position. The distance between the two marks is eight centimetres.
Align the two two centimetre indications of the tool with the X axis.
Draw the contour line for the sine from the X value of 0. Start at the pen blocker to avoid glitches. Draw along the contour until the pen blocker at the X value of 180 degrees which is on the X axis. Hold the pen on this position and rotate the tool around this pen position so the tool is upside down. Continue drawing the second half of the tool until reaching the pen blocker again on the X axis.
For convenience place a pushpin at the 180 X position and rotate the tool around the pushpin.
You can also place pushpin(s) in the pushpin marker positions to hold the tool in place.
Cosine graph
Photo: The first quarter of the cosine graph is drawn. The tool is rotated 180 degrees clockwise, so the ‘hill’ is upside down for the negative part of the graph.
Basically the shape of the cosine has the similar shape as the sine. However the graph is shifted to the left for 90 degrees.
Draw an X axis with the ruler. Mark the 0, 90 and 180 degrees positions. A section of 90 degrees has a length of four centimetres.
Align the tool with the X axis where the right pen blocker is positioned at the 90 degrees position. Draw along the contour starting at the top (X=0) downwards to the X axis.
Rotate the tool 180 degrees clockwise. Align the tool with the X axis and draw the negative part of the graph from 90 to 270 degrees.
Rotate the tool once again. Align the tool with the X axis and draw the last quarter of the graph up from the X axis on to the top of the graph.
Tangent graph
Photo: The X and Y axis are drawn as well as the first quarter of the tangent graph. The tool is positioned so the second quarter can be created. The pen blocker is hooked with a pushpin.
Draw an X axis with the ruler. Mark the 0, 90, 180, 270 and 360 degrees positions. The distance between each position is four centimetres, which covers 90 degrees.
Draw two asymptote lines : One at 90 degrees and one at 270 degrees.
The distance for the first asymptote line is four centimetres from the 0 degrees position on the X axis, the next one is at a 12 centimetres distance.
For the first section of the graph align the short (four centimetres) side with the X axis. Start at the pen blocker (0 degrees) and draw along the tangent contour upwards until you approach the first Y axis.
For the second section of the graph find the 180 degrees position (preferably provided with a pushpin).
Hook the pen blocker with the pushpin while the tangent curve is pointing down. Align the tool with X axis and draw the lower part of the graph.
For the third section rotate the tool around the pushpin 180 degrees counter clockwise. The tool has the same position as for the first section.
The fourth section is similar to the second section. Find the 360 degrees position, hook the pen blocker to the pushpin and align the tool with the X axis and draw the curve going down unstill you approach the Y axis.
Parabola manual
Photo: A parabola has been drawn at X=0 and Y=0. The tool is positioned to create a second one at X-3 and Y=2.
Draw an X and Y axis.
Align the bottom indent with the Y axis and the ones in the poles with the X axis.
Draw along the contour of the tool.
You may choose any other coordinate as the minimum for the y value. The centimetre indications help to align the tool with the axes.
Lens manual
Photo: for the application the tool is rotated 90 degrees. The median light beam is drawn. A convex lens has been drawn. The tool is positioned to create the mirror.
Draw a line with the ruler to represent the median light beam.
Align the tool on the line.
Draw the inner contour for the mirror, the convex or the concave lens.
You can add additional light beams to show light breaking and reflection effects.
Hyperbole manual
Photo: An X and Y axis are drawn. The graph is drawn in the first quadrant. The tool has been rotated 180 degrees clockwise so the second part can be created. A pushpin is at position X=0,Y=0.
Draw an X and Y axis with the ruler
Be aware, the effect of the additional material is also that the actual X and Y axis are covered by the tool.
Align the tool with the two axis by placing the flat sections at the tails with the axis.
For convenience you can place a pushpin in the crossing of the X and Y axis.
Draw along the contour of the tool.
Rotate the tool 180 degrees clockwise. Align the tool with the axis and draw the second part of the graph.
Preparation
The default setup has rubber bands every two centimetre in horizontal and vertical direction, so forming a grid.
Two rubber bands go around the frame to indicate two axes.
However, before mounting the GraphGrid frame on the TactiPad make sure all rubber bands are provided according your preference.
Low and high rubber bands – Using differences in height
The framework of the GraphGrid has a thickness of four millimeters. Therefore there is a well noticable difference in height between the bands that run along the top side (outer) part or the bottom side (inner)
Part of the frame. The lower rubber bands that are attached to the hooks run alongside the inner part of the GraphGrid and can lay flat on the drawing surface.
The rubber bands can be placed horizontally and vertically with a minimum distance of 1 centimeter.
The bands can also be placed at an anglle of any number of degrees.
Note: The hooks at the inside edge are at every centimetre. Adding rubber bands to each hook will result in a (too) denced grid.
Placing the grid lines and axes
The hooks and indents are positioned every centimeter along the frame, so the minimum cell dimensions are one by one centimeter. To form a larger grid, you can place the rubber bands two or more centimeters apart by skipping one or more hooks.
Spare rubber bands
Additional rubber bands for more grid lines and axes are supplied with the GraphGrid. These are standard, thin rubber bands with a length of 12 to 15 centimetres.
To keep the rubber bands in place, the hooks have such a shape that the rubber bands will not come loose if they are positioned properly in the frame. Pplacing a rubber band is easiest if you hold the rubber band with two hands, keeping it perpendicular to the frame so you can slide it through the slot to the end of the hook. First of all, you span the rubber band in two opposing hooks, so that it forms a double line between two sides of the GraphGrid. You then take the upper of these two lines and slide it into the next desired hook in both sides of the frame to form the second grid line.
The higher rubber bands are stretched around the outside of the GraphGrid and fall naturally into the indents.
Getting started
To begin with, make sure the TactiPad has the landscape orientation with the hinge facing backwards.
To hold the GraphGrid frame,, place the knobs at aproximately five centimeters away from the corners.
To position the GraphGrid frame correct, place the flat corner in the upper right corner of the TactiPad.
It is possible that of one of the rubber bands coincides with the position of one of the knobs. Therefore, the holes in the GraphGrid around the knobs are extra-long so you can slide the knobs aside for the desired placement of the rubber band.
The TactiForma set consists of the following tools: 7 squares, 7 triangles, 1 rectangular L-shape hook, 5 polygon tools plus magnetic foot, 2 spur wheels, 4 ellipses.
Selecting a tool
The square tool, the equilateral triangle tool, the ellipse tool and the regular polygon tool come in different sizes. Knowing about the width of one, two or three of your fingers or the length of one finger, makes it easier to measure distances or finding the right tool without (extensive) counting the measurement indications.
Although the tools are recognizable by their shapes and the names of the tools clarify their purposes, it is highly recommended to read the descriptions and manuals because the tools contain many more applications than you will notice at first sight. Especially the regular polygon deserves reading the detailed description and manual while the usage differs from the rest.
Shape or tool
To avoid confusion while reading the texts where the name of the tool and the shape that is created is used, the words shape or tool will be added as much as possible.
Terminologies
We introduce some of our own terminologies that are applied throughout the pages such as finger fitters, finger entrance, small, medium and large sized indents for measuring and alignment.
The frame that forms the square is one centimetre wide. Two finger fitters are provided in opposing sides. Two opposite diagonal corners are rounded. The others are sharp. At each corner a pushpin marker is located. Along the outside and inside a small indent is provided at every centimetre. In case the length of the side is odd numbered, there is in additional indent at the halfway position.
When you position the square somewhere on the TactiPad in any orientation and then draw along the inner contour, ou create your first square. Alignment for horizontal or vertical oriented shapes. Aligning the tool with the ruler that is placed on two opposing TactiPad knobs, will make sure the aligned side is horizontal or vertical, relative to the measurement indications of the TactiPad.
Other shapes
With the square tool you can create many more shapes such as a rectangle, triangle, parallelogram, trapezium and also 3D figures like cube or pyramid.
Rectangle 1
Choose the square tool that has the longest side of the rectangle that you want to create. Draw the bottom line. Determine the width (is height) of the rectangle and draw the left and right line from the same position on to the bottom line. Move the tool downwards so the top side is almost on the line endings of the two (vertical) sides. For convenience place a pushpin in the positions along the two short sides and draw a line from here to the bottom line. Move the tool towards you until the top side hits the two pushpins. Draw the top line and remove the tool.
Rectangle 2
For larger rectangles alignment with the ruler works well. Place the ruler on two knobs. Place the tool against the ruler and draw the bottom line and the left side. Move the tool to the right for as long as you want the length of the bottom line of the rectangle. Draw the rest of the bottom line and also the right side of the rectangle. Connect the two line endings of the sides with the ruler.
Photo: Square against ruler, which is placed on two knobs, moved to the right after drawing the bottom line and left side.
45 triangle – 1/1/sqrt2 triangle
By drawing two connected full sides of the square and the diagonal line between the two line endings will create a rectangle triangle, also known as a 1/1/sqrt2 triangle. Placing pushpins in two diagonal positioned corners can help to draw the diagonal line.
Right corner triangle
By drawing only parts of two connected sides from the tool and connecting the line endings, will result in a right corner triangle.
Triangle with sides 1/2/SQrt5
Draw a line from a halfway position into a corner. Continue with a line along the full side. Remove the tool. Connect the two line endings. This is also a right corner triangle with specific proportions for the length of the sides: 1/2/SQrt5.
Isosceles triangle
Calculating the surface of an isosceles triangle created with a square tool is not too complicated. Draw a line as the base line for the triangle and check the halfway position on this line. Mark or place a pushpin in exact the same position in the opposite side. Connect this top corner position with each of the two line endings of the base line. The surface is the multiplication of the length of the base line and the length of the side of the tool that is used, divided by two.
Irregular triangle
By drawing – a part of – a side and then marking or placing a pushpin along one of the three other sides and providing lines from this position on to both line endings will show an irregular triangle.
Parallelogram
Draw the bottom side of the square.
Place a pushpin two or three centimetres to the right in the bottom line.
Move the tool to the right until it hits the pushpin. Make sure the tool is still aligned with the bottom line.
Place a pushpin at the desired height position at the inner left side of the tool. Mark or place a pushpin in the right side on the same height as well.
Move the tool downwards until it hits the pushpins. Draw the full top side of the square/parallelogram.
Remove the tool and connect the lines.
Trapezium
Draw the bottom line of the square. Use a smaller square tool as before. Align the bottom line with the one that was created already. Draw the top line of the smaller square. Connect the line endings between the top and bottom line. When placing the halfway position of the smaller square at the halfway position of the larger square, the trapezium will be symmetrical.
Cube 1
Draw a horizontal line on the drawing board and leave the ruler sitting on the knobs.
Align the square tool with the ruler. This square is the front face of the cube.
Draw a second square, the rear face, a little to the right and a little upwards. Make sure the bottom lines of the squares run parallel. Therefor move the ruler up a little.
Connect each of the four corners with the ones closest of the second square.
As the above is a start, you have to be aware that some of the lines are invisible. They should be dashed as in the below example
Cube 2, top right view
The ‘top right view’ of the cube shows the front face, the right face and the top face of the cube. Each corner is given a character indication. The bottom face of the cube has corners A, B, C and D. The top surface has corners E, F, G and H. Corner E is above corner A. Please note, there are also other naming conventions.
Steps
Step 1, front face: Draw the inner contour of the square. The lower left corner of the front face square is called A, the one at the lower right B, going straight up from here is F and going left is corner E.
Step 2, rear face: Move the square tool a little to the right and a little up (*). Make sure the lower side is parallel to the lower side of the front face.
The rear face has corner D in the lower left and going counter clockwise C, G and H.
The square of the rear face is partly covered by the front face. So two of the sides of this square have to be dashed: C-D and D-H.
Step 3, connecting corners: Four lines to connect the corners of the two squares complete the cube. Only the line A-D has to be dashed, the others are solid.
(*) Little to the right and a little up
About the amount for ‘a little to the right and a little upwards’ can be discussed. A rule of thumb is to imagine a line under an angle of 30 degrees starting in the A corner. Set out a distance of 60/70% of the side length along this line. The hypotenuse of the 30triangle can also help here.
Pyramid (symmetric)
For a start take the eight centimetres square and equilateral triangle with the same side length. The base of this pyramid is drawn as a parallelogram. The four side faces are triangles. The front face is equilateral.
Draw a parallelogram where the top side is moved upwards and to the right with the rule of thumb in mind; an angle of 30 degrees and a length of 60% of the length of the side.
Dash the left hand side and the top side of the parallelogram because they are invisible.
Place the triangle aligned with the bottom line of the parallelogram where the corners match. Draw the two remaining sides of the triangle where they meet in the top corner T.
The corners of the base (floor) of the pyramid are named A, to D, counter clock wise.
To complete the pyramid, provide a solid line from C to T and a dashed line from C to T.
Taller pyramid
The top of the pyramid can be placed in many positions and the figure will still be a pyramid, but not symmetrical. To keep it as such, the top T has to stay on the altitude line that arises from the crossing of the diagonals; the lines A-C and B-D, the centre of the parallelogram. To set the top T, use the ruler that goes across the TactiPad, in equal positions of the frame and also meeting the centre position of the parallelogram.
The body of the triangle is about 12 millimetres wide. On all three of the outer sides a finger fitter is provided for easy lifting or extra grip. One outer corner is rounded, the other two are sharp. All three corners have a pushpin marker at the top surface. Along the outside and inside are indents located at every centimetre. They align with each other. In case the length of the side is odd numbered, there is an indent in addition at the halfway position.
Remarks
The TactiForma triangle tools are designed as equilateral triangles. This shape is more difficult to draw than an irregular triangle; therefor, you can just mark three positions or place three pushpins somewhere in the drawing area and connect them with a straight line. More triangle variations with specific properties, can be created with the square tool or the rectangle hook tool.
When you place the triangle tool somewhere on the TactiPad in any orientation and then draw along the inner contour, you have created your first equilateral triangle.
Right corner triangle
Mark with a dot, or for convenience place a pushpin at the halfway position of a side. Start to draw a line from here into a corner. Continue drawing a full side from the same corner. Draw a line from the halfway position to the opposite corner. You have created a 30/60/90 degrees triangle, also known as a 1/2/square root3 triangle.
Isosceles triangle
Determine what is going to be the bottom side of the triangle. Draw the bottom line from the desired corner position on to the halfway position. Continue this line until you reach the position that has the same distance to the halfway position as your starting point. The starting point is not necessary the corner of the tool.
For convenience you may place pushpins at the corner positions first or after you have created the base line. Just mark the top corner or place a pushpin in the top corner and provide the lines from the top corner to both base line corners.
Irregular triangle
Draw a section of one side, not necessary starting from or ending at a corner of the tool. For convenience you can place pushpins first.
Mark or place a pushpin in the desired position along one of the other sides.
Provide lines from the marked position to both line endings.
Diamond with 60 degrees angle
Place the triangle on the TactiPad. Draw two of the three sides fully. For convenience place a pushpin in one of the line endings. Rotate it 180 degrees around this pushpin.
Draw a line from here into the corner. Draw the last line, so they connect again at the line ending.
Diagonals can be provided by drawing the lines between opposing corners.
Photo: The triangle is rotated over 180 degrees, to draw the second part of the diamond.
Draw a house
For the outline of a traditional house use a square and draw three-quarter of the shape.
Leave the top side open.
To draw the roof, use a triangle, preferably a little larger than the square size.
Place this above the walls of the square and draw the inner contour as the top part of the front face.
Use the smaller square tools to add windows.
Fantasy
Draw a triangle and place a pushpin in one of the corners. Rotate the triangle in any of both directions and draw again – a part of – the triangle. Repeat this a few times and change the corner to rotate.
Four equal sided pyramid
To start, for convenience, use one of the larger sized triangle tools. Orient the bottom side horizontally. Draw the inner contour.
The rectangle hook tool has the shape of a capital l, where the horizontal and vertical part of the character have the same length of 10 centimetres / approx. 4 inches. The two parts enclose an angle of 90 degrees. They are equipped as a ruler with measurement indications in centimetres at the top surface and indents per half centimetre along the sides. The body of the tool is two centimetres wide. On the outer side, near to the rounded corner and to the tips you find finger fitters for easy lifting or extra grip. The corner between the sides is rounded at the outer side. At the top surface is a groove under an angle of 45 degrees for alignment. The rulers are ending with a 90 degree hook serving as a pen blocker. Pushpin markers are located near the rounded corner and near to the tips.
By drawing along the rulers into the enclosed corner, two lines under an angle of 90 degrees occur.
Right corner triangle
When connecting the two endings of the previously created lines you get a right corner triangle.
Diamond
For the centre position, use a pushpin. For high precision you might want to use four additional pushpins for the corners.
Place a first pushpin at the centre position of the diamond.
Place the enclosed corner of the rectangle hook against the pushpin in the desired orientation.
Most logical is to have one ruler pointing left, the other ruler downwards.
Mark or place a pushpin at the desired horizontal ruler position for corner A.
Mark or place the third one along the vertical ruler for the B corner.
Now The marked positions or the three pushpins are positioned in a triangle.
Lift the tool and rotate 90 degrees counter clockwise.
Place the enclosed corner of the tool again against the pushpin that marks the centre position and make sure the ruler touches the B pushpin.
Place a fourth pushpin for corner C, which is opposite A.
Rotate again 90 degrees.
Place a fifth pushpin for corner D in the similar ruler position as for corner B.
Connect the four corners.
Kite
In the kite shape the distance from one pair of opposing corners to the centre position differs. In other words, one of the diagonal lines is cut in two sections with different lengths. Apart from this, the drawing instructions are similar as for the diamond.
Photo: Creating the last line of the kite conform the instructions.
Challenge
Fit a diamond exactly in the eight times six centimetres ellipse. The corners of the diamond are all four exactly on the ellipse outline.
The body of the tool is symmetrical. It can be seen as a circle, equally stretched in 2 opposite directions. The frame of the tool is one centimetre wide. At the outer side the frame has 2 finger fitters for easy grip. Pushpin markers are located at 0, 90, 180 and 270 degree angles relative to the centre position. So the markers indicate the longest and shortest diagonal of the tool. As the TactiForma ellipse is a circle based shape, radii can be drawn from the centre. Indents per 10 degrees along the outside indicate the angles.
Place the ellipse tool on the TactiPad and hold it at the finger fitters positions. Draw along the inner contour. Finding the centre position.
As long as you create two crossing lines that are symmetrical, they will pass the centre position. Symmetrical here means each line is a diameter line, from opposing angle positions, not necessary the lines of the longest and shortest diagonal. When connecting the pushpins at 0 and 180 and also those at 90 and 270 degrees, you obtain the longest and shortest (orthogonal) diameters. The lines will cross the centre position.
Be aware, when placing all four pushpins at the outside and then lifting the tool might cause the pushpins fall out. To avoid this, place two pushpins at the inner side a 0 and 90 degrees and two along the outside at 180 and 270 degrees. By moving the tool slightly upwards off from the pushpins you can safely lift the tool.
Photo: Ellipse with four pushpins, two at the inner side at position 0 and 90 degrees and two at the outer side at positions 180 and 270 degrees.
Placing the ellipse on a pre-determined centre position
The open space at the inside of the ellipse does not make it easy to place the ellipse centre on a pre-defined position. You have to create support lines.
Dome
Instead of drawing the full contour you can draw parts or a dashed section of the ellipse. Dashed lines are applied to indicate invisible lines of 3D figures. For a dome, orient the longest diagonal of the ellipse horizontally on the TactiPad. Draw the lower part of the contour as a solid line and the top half as a dashed line. Now you have the bottom section of a dome. The dashed section represents the invisible part of the dome.
This 3D figure can be completed by placing the top (T) above the centre and dashed section of the contour and providing two lines from the position T to both ends of the virtual diagonal. While the push pins are still there, the line for the diagonal at the bottom should be dashed as well because this line is not visible.
Photo: Baseline with pushpin on halfway and opposite position, which is connected with the line endings to create ta dome.
Cylinder
The cylinder has two ellipses; one for the bottom and another for the top side. Draw the bottom as for the dome. Move the ellipse upwards. Make sure the ellipse for the top surface does not cros anywhere the bottom side ellipse. Connect the two ellipses with lines at the horizontal diagonal positions.
Fantasy 1
Create an ellipse and place a pushpin somewhere along the inner side and rotate the ellipse and draw it again. Choose a different position for the pushpin. Repeat this a few times, for a unexpected result.
Fantasy 2
Combine the ellipse with the triangle or square and fit the shapes into each other.
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