What are angle, sin and cosine ? 
Oct 8th 2018

Angle and related values are always defined with triangle and circle. Just like in the picture on the right. It is hardly ever said and very poorly known, that angle is actually a line with common x and y stepping. Angle's numeric value is an alternative way to present x and y multipliers for the line.

Angle misses C from base formula for lines ( Ax + By + C = 0), in resolved formula b is missing ( y = kx + b, where k = A/B and b = C/B  )  The line for angle goes always through origin. 
- Angle 30 : 2x + y = 0, Angle 45 : x + y = 0, and so on.
- Cotangent is k for line ( k = x/y ). It is resolved with Y = 1, Cosine is another k for the line. Cosine is resolved, when line length is 1.
- Tangent and sin are unique to angle. They are y-axis projections. Tangent is resolved with X = 1, sin is Y axis projection for line length 1.

When you go further, in angle line's x-y components are embedded into ratio.  Numeric angle is always a simple ratio for dividing x and y. Regardless of the used unit :
- When angle is 30 : x = 2, y = 1 ( take one step right, then one up-right step, take step right, one up-right again, ... )
- When angle is 45 : x = y ( take up-right steps all the time )
- When angle is 60 : x = 1, y = 2
- When angle is 90 : x = 0, y = 1
- When angle is 0 : x = 1, y = 0
- When angle is 91 - 179 : y > 0, x < 0 =|= 181 - 269 : y < 0, x < 0  =|= 271 - 359 : y < 0, x > 0

Nowadays when you  have computers you do not need sin, cosine, etc. tables. You can convert angle into line's x and y components. With x-y components you can calculate sin, cosine, tangent and cotangent values. When you divide angle into x-y components, you can calculate angled values without sin and cosine. When you need extreme accuracy, you can clamp the values so, that you can use faster and more accurate Int64 variables in your calculations.

Normal for the side of triangle
Oct 2018

Length of parallel / normal for side goes with ratio of the surrounding angles.

You cannot lead the formula from the above

In imaginary triangle, where all angles are 0, the angle is the length of side :
- when angle is zero, the length of side A is C₁ and B is C₂
- Sin clause creates 1 to 1 connection in between angle and length of side.
- therefore the length must always be the same than ratio of angles.

You need it in for example calculating elliptical lines. Searching the cross point for tangents. And lines in general.

When alfa or beta is over 90 degrees, you must throw the parallel outside C ( extension of C ) , when needed.

In school got 6 from calculus, scale was 10 to 4, where four is failure, seven average.  In my reports physics and chemistry varied at the same low levels. Grades from automotive related things were like phone numbers in computer games : 555-555555 . . .

Elliptical line
Oct 2018

Elliptical line allows you to easily draw and create rounded shapes.

By default line has three points. Start point, end point and cross point of tangents for drawing the ellipse.
- When line begins or ends the shape you need points for giving the direction to tangent.

Line makes always a smooth rounding to the corner. When you change the angles of the lines, the rounding changes accordingly.

In computers you can draw the line with bezier curves. In 3d modeling the line allows you to use rounded walls and floors. You can never create impossible shapes with the line. Currently all rounded shapes are made with short lines and shaded textures.

- Ellipse is fully described / unique when it has three points. A, B, C points defines the ellipse.
- A and B points defines also the segment for ellipse. C point defines the direction / side for the segment.

How to calculate and draw spiral / Archimedes loop ? 
Oct 19th 2018

Archimedes loop is spiral, where distance from outer and inner line is equal. There is fixed distance in between each line.
In at least my books, the system with what you calculate and draw Archimedes loops was left blank. 

The system is simple
The spiral is built with 180 degree arcs.
You have a set of overlying arcs, then you have a set of underlying arcs.
Center points of both sets have shared y coordinate. But sets have different x coordinate. 

As you can see from the picture, the shift-x in between sets center points is delta R per 2. 
- Delta R is the amount with what you reduce or increase each round.
- In drawing tools delta R seems to be somewhere near one pixel ( Corel Photo Paint - Rotational Blur, etc ).

Archimedes loop is the most useful from spirals. Besides drawing, it is easy to construct. To real life construction, CAD design or 3D world.

180 degree arc is native to Archimedes loop. When you use 90, 60, 45,30 degree arcs in the construction. It results into space in between the rounds. All rounds need outer and inner lines / walls. The center points of the upper and lower sets are not in the same place.

You cannot use the formula for another popular spiral : hypnotizers Vertigo spiral. In Vertigo line width grows all the time. Delta-R, white space in between black stripes  is usually steady. Sometimes it grows with line width. More difficult Vertigo can be tailored with under and overlying sets of ellipses.

- spiral is nice looking thing for decorative purposes. Spiral is used in for example some standard hair dressings for women.
- One potential usage for Archimedes loop is lead-in to and lead-out from rounded system. If you for example use heavy trucks for filling a round system, you cannot simply drive over the main loop. You need lead-in, which lies on solid ground. Lead out is another possible usage. Stiff, almost solid liquid needs round tracks for steady flow. It slows down and gets stuck in sharper corners.
- Coil for car uses often two spirals ( shrinks towards both ends ) for creating progression and increasing the length of maximum suspension event. With spiral you can create rounded telescopes. 
- Spirals are not often seen in our world. One possible reason is the earlier mentioned blank space in the book of magic. We'll see. 

Failed to find resolved mathematical formula for calculating cross points for two lines. Line's resolved formula is y = k * X + B. 

When you calculate cross points, you have two formulas : Y1 = k1 * X1 + B1 and Y2 = k2 * X2 + B2.

These equations collects into following :  k1 * X1 + B1 = k2 * X2 + B2. ( Y1 = Y2 )

Since X1 and X2 are also the same, you can go into  : (k1 - k2) * X = B2 - B1

From this the final formula goes into  X = ( B2 -  B1) / ( k1 - k2 )

After you get X, you get Y by placing X into either y = kX + B formula.

Checked the calculation graphically with CAD. In my old math books, messy matrix for this simple calculation was footnoted, that it might be possible to find a way to calculate the point without iteration.

Lines angular factor, which is marked as k, is merely a multiplier for Y coordinate. When X grows with one unit, Y value for the new X comes with angular factor. When you multiply X with angular factor, you get Y coordinate.  X comes from Y with division. Also possible to calculate k_x factor for the line. You get angular factor with two known points in the line. You calculate both X and Y lengths in between two points. After you get the distances ( simple subtraction ), you divide Y-length with X-length. k_x factor comes, when you divide X-length with Y length.

How do you calculate points distance from known line ?
You sort of build parallel line from the known line. This parallel line goes through the calculated point. After this you resolve distance with Pythagoras application. You use X and Y multipliers as sides. Mathematic Hesse has puzzled out this general ABC formula. You can calculate distance also with Pythagoras, known start and end points of the line.

You can "unpack" angular factor into A, B, C with earlier k = dY / dX and two randomly selected points. Since k is also k = A/B, with assigned points you must get some numeric values to general formula's A and B. When you have b and B, you can resolve C from b = C/B. A's and B's numeric values are insignificant, they form ratio in between x and y axis growth. C shifts line from origin. General formula doesn't have explicit coordinates for y axis crossing.

No idea, who has created that messy ABC system. When Ax + By + C = 0, line has one point, where placed x and y values results into formulas 0. In unpacking, you don't have to worry about mysterious, fixed X-Y point. In selected computer graphics you use Ax and By stepping system for matching analog points into digital pixels.

Number series with number 1 as base
Mar 13th 2020

In atmospheric pressure the base pressure is 1 bar. Pressures from one to infinity are over pressures. Pressures in between 0 and 1 are under pressures. Scaling ... enlarging and shrinking ... is another thing, which use one as base number.  Both under and overpressure are linear series. These series can be presented in two ways.  There is linear and progressive presentation. Without conversion over pressure comes with linear presentation and under pressure with progressive presentation. 

The formula with what you can convert the series is 1/X and -1/X. 
- When you create negative number series for under pressure -2 comes when you make division 1 / 0.5 and multiply the result with -1. When you continue 1/0.3 = 3 and 1/0.25 = 4.
- When you create progressive series from values over 1, you make division 1/2 = 0.5, 1/3 = 0.33 and so on.
- You can also use other base numbers than one. When you do so, you replace the fixed number one with the base you want to use.

In Liquid Fuel engine and other force-power calculations or estimations, you the conversion gives you comparable over and under pressures. -20 bar vacuum comes with 5% filling, this vacuum has and needs as much power as 20 bar pressure. -1000 bar vacuum comes with 0.1% filling and so on.

In general usage liquid fuel engine's filling ratios are measured pressures. So that -20 bar under pressure is a match to measured 0.05 bar pressure.

In weather forecasts and hurricane-typhoon under pressures, scaling gives you comparable low and high pressures.

Diamond
Mar 20th 2020

I was making calculation sheets for rolling engine. Piston-wheel for rolling engine has rounded bottom. Piston bottom's diameter comes cylinder. Required depth of rounding is bound to diameter. When I checked the books, there wasn't resolved system for round bump or pit, whose diameter and depth is known. Spent one lazy working day with the system. Calculation is simple application from Pythagoras and sin clauses. Base dimension for the pit or bump is calculated with diamond like shape. 

The calculation and geometric proof

In the blue sector you see base situation. The sample pits diameter is 10 cm, and depth of pit is set to 2.5 cm.
- With these two values you get length for hypotenuse and angle from surface to the bottom and center of the pit.

In reddish sector on the right you discover a matching straight angled triangle for the pit triangle. With this discovery get you diameter for the pit. 
- In matching triangle earlier hypotenuse converts into side and hypotenuse converts into diameter for the pit.
- Angle system goes upside down with rules for crossing lines. The source triangle's 26.57 degrees must go to the other at top of the pit. When so top angle must be 63.43 degrees. This moves 26.57 degrees to the top of the bigger triangle ( sum of triangle's angles must be 180 ).

You can calculate diameter with angle or ratio of the pit triangle sides.

In green sector on the left you see proof for the system. 
- there is symmetric rectangle.
- Rectangle's diagonal goes through center point and origin for pit circle.

Calculation is a match for drawn and measured system.

- - - -

With calculation it is easy to add bumps and pits into products and 3D models.

In drawing tools you can have a function / tool for adding and modifying pits and bumps.

Elliptical balls - volume from two areas
Mar 20th 2020

Volume of the ball is width * height * depth, just it is with boxed volume.
- The difference is, that you calculate two areas for the rounded shape. Ball is simple it is 3.14 * r³. 
- In order to calculate volume for elliptical you need formula for calculating volume with two areas.
- When you calculate two areas for the box, both areas have one shared side. 
- When you multiply areas together and divide the result with shared side, you get volume with areas.

Since area's calculation system for boxes and balls are the same, this aerial volume calculation should be valid balls and ellipses.
When you play with balls, you must remove radius and pi from the result. Volume calculation has 3 parameters, but aerial multiplication has four parameters. When so 

V = (a * b) * (b * c) / b ==> V = a * b * c

V = (3.14 * R1 * R2) * (3.14 * R2 * R3) / (3.14 * R2)
- formula simplifies into form V = 3.14 * R1 * R2 * R3

The aerial volume calculation should work as long as you have one shared side for two areas. You must always be capable of removing one known and shared side from result. 

Electric 3D pulse and 3D snapshots

Media players analog hard disk allows you store 3D data. Electric 3D track needs variable track width. On 3D track you store the variable charges heights onto track widths. You move the recorder-reader head  sideways and lengthwise. From this 3D data you can get volume by reading and summing electric charges from the track area.

In more advanced 3D storage, you use wider head, which record-reads the track width at once. Recording and playback must support the recording system. System  where you sum currents-voltages from disk-track areas is possible also with this system.

With the system you can take 3D snapshots / photos. Then you can transmit live 3D feed. It is not possible to store 3D feed onto 2D surface. You cannot expand analog 3D onto 2D without slicing the feed into stripes. Since we live in time, time is not available for our self made systems.

Recording of 3D feed needs a wire, where you put the feed to the end of the wire and push the earlier recordings farther from the end. During playback you move the recording in the wire to other direction. After reading the pushed data, the reader must push the incoming 3D feed to another wire. Otherwise the recording can be read only once. x-Ray reader is also possible. In x-Ray system, where the reader penetrates into recording, you bring the surface to recorder as liquid. In the recorder you have continuous process, which cools liquid into solid and records the feed.

 In current wave theory, both ways of recording 3D feed are not impossible, but very difficult and challenging. 3D snapshots are possible almost right away.

What is pee ? 
Aug 28th 2019

What is Pi ?
Pi is constant, with what you can calculate line length for symmetrical half circle. From known distance A to B.
Pi has another usage in the calculation of circle's area. With Pi you can get the area, full circle fills from squared rectangle.

What is half circle ?
Half circle is special, simple form of parabolic curve. In half circle X and Y grows evenly. It results to parabolic curve, that has end point.  In common parabolic curve y grows less than x.
When half circle is parabolic curve, it is possible to extend parabolic curve from half circle to a crossing parabolic curve. Best known crossing parabolic curve is first Christians sign, symbolic fish. 
Drop is 3 dimensional crossing parabolic volume.

What is ellipse ?
Ellipse is a general name for rounded, closed shapes.
- In math and computers ellipse is symmetrical shape. It is built with two arcs, which are connected together. The angles at the end point matches.
- Parabolic, symmetrical ellipse is built with two parabolic curves. At the crossing point both parabolic curves have the same direction.

Third and fourth ellipses are asymmetrical ellipses. Asymmetrical ellipse has different rounding at the ends.
- Mathematical, asymmetric ellipse needs three arcs.
- Parabolic, asymmetrical ellipse needs three parables.

Fifth and sixth ellipses have different looks and rounding at all four turn points. It needs four arcs or parables.
- this form doesn't use bounding 90*90*90*90 degrees rectangle.

We cannot calculate parabolic ellipses. Free hand and iteration are the methods for drawing them.
Parabolic rounding might have some usage. In for example creation of aerodynamic shapes. You could get an optimal rounding to raise from grille to hood.
First you have sharply rounded grille from which you raise smoothly to slightly tilted hood. Parable is ideal shape for the purposes of this kind.