Implementing ATAN2

An implementation of ATAN2 in PHP. PHP implements ATAN2 but I needed to figure out the algorithm for a microcontroller that didn't, so I prototyped it in PHP.

Here is the code:

class Math {
  public static function atan2( $dy, $dx ) {
    if( $dy > 0 ) {
      if( $dx > 0  ) $tcl = atan($dy/$dx);
      if( $dx < 0  ) $tcl = M_PI - atan(-$dy/$dx);
      if( $dx == 0 ) $tcl = M_PI/2;
    }

    if( $dy < 0 ) {
      if( $dx > 0  ) $tcl = -atan(-$dy/$dx);
      if( $dx < 0  ) $tcl = atan($dy/$dx) - M_PI;
      if( $dx == 0 ) $tcl = -M_PI/2;
    }

    if( $dy == 0 ) {
      if( $dx > 0  ) $tcl = 0.0;
      if( $dx < 0  ) $tcl = M_PI;
      if( $dx == 0 ) $tcl = 0.0; // the 2 points are the same, default to zero
    }

    return $tcl;
  }
}
And here is an example of how to use it:
$heading = 270.0;  // degrees
$Ax =  0.00;
$Ay =  0.00;
$Bx = -7.89;
$By = -0.86;

// get target distance and bearing in degrees
$dx = $Bx-$Ax;
$dy = $By-$Ay;
$dist = sqrt( pow($dx,2) + pow($dy,2) );
$dir = Math::atan2( $dy, $dx ) * 180 / M_PI;
printf( "Target spotted at %0.3f degrees, distance %0.3f !!\n", $dir, $dist );

// get new heading
$diff = $heading - $dir;
if( $diff < -180 ) $diff += 360;
if( $diff > 180 ) $diff -= 360;
printf( "Turn %s %0.3f degrees\n", ($diff < 0 ? 'left' : 'right'), $diff );

Implementing Logic Gates in a Spreadsheet

Here are some of the basic logic gates, implemented as spreadsheet formulas.

GateSpreadsheet Formula
ANDAND(A,B)
OROR(A,B)
NOTNOT(B)
NANDNOT(AND(A,B))
NORNOT(OR(A,B))
XOROR(AND(A,NOT(B)),AND(NOT(A),B))
XNORNOT(OR(NOT(OR(A,NOT(OR(A,B)))),NOT(OR(B,NOT(OR(A,B))))))

It is possible to implement any gate using only NOT and OR (NOR), or with NOT and AND (NAND) gates. For example, here is an AND gate implemented using only NOT and OR gates:

AND using only NOR gatesNOT(OR(NOT(OR(A,A)),NOT(OR(B,B)))

With AND, OR and NOT, we can implement a half adder, like this:

Column AColumn B
A0
B1
Carry (AND)=AND(B1,B2)
Sum (XOR)=OR(AND(B1,NOT(B2)),AND(NOT(B1),B2))

By string together two half adders and OR-ing the carry bit, we can create a full adder:

BitSpreadsheet Formula
A0
B0
Cin0
Cout=AND(B2,B3)
Sum=OR(AND(B2,NOT(B3)),AND(NOT(B2),B3))
Cout=AND(B6,B4)
Sum=OR(AND(B4,NOT(B6)),AND(NOT(B4),B6))
Cout=OR(B7,B5)
Sum=B8

Implementing a 4-bit adder is just a matter of simply replicating the full adder logic four times:

8421
Input A0000
Input B0000
A=B2=C2=D2=E2
B=B3=C3=D3=E3
Cin=C12=D12=E120
Cout=AND(B5,B6)=AND(C5,C6)=AND(D5,D6)=AND(E5,E6)
Sum=OR(AND(B5,NOT(B6)),AND(NOT(B5),B6))=OR(AND(C5,NOT(C6)),AND(NOT(C5),C6))=OR(AND(D5,NOT(D6)),AND(NOT(D5),D6))=OR(AND(E5,NOT(E6)),AND(NOT(E5),E6))
Cout=AND(B7,B9)=AND(C7,C9)=AND(D7,D9)=AND(E7,E9)
Sum=OR(AND(B7,NOT(B9)),AND(NOT(B7),B9))=OR(AND(C7,NOT(C9)),AND(NOT(C7),C9))=OR(AND(D7,NOT(D9)),AND(NOT(D7),D9))=OR(AND(E7,NOT(E9)),AND(NOT(E7),E9))
Cout=OR(B8,B10)=OR(C8,C10)=OR(D8,D10)=OR(E8,E10)
Sum=B11=C11=D11=E11

Radians made easy

If you are in danger of losing your Geek cred because you don't understand radians, fear not! Radians aren't bad, they're just drawn that way.


What is a radian?

Diagram A
You already know some things about degrees, like there are 360 degrees in a circle and that you can measure an angle with two points on the circle's edge (see Diagram A).

Radians are just another way to measure angles. Here's how:

Start with a circle. The radius of the circle is a line from the center of the circle to a point on the edge. In Diagram A, that's the line CA. Start point A moving counterclockwise around the circle. It will trace out an arc as it travels. When the arc is as long as the radius of the circle, stop. Call that point B. You now have an arc AB, and an angle ACB. That angle at C is equivalent to one radian. So, the length of arc AB is the same as the length of the radius of the circle, which is line CA. Here's the rule:
  • The lengths of the arc and the radius are equal.
Next, you already know that the ratio of the circumference of a circle to its diameter is pi, or π, which is 3.14. The formula is C = πd. Since the radius of a circle is half the diameter, then d = 2r, so C = πd = π2r, more commonly written C = 2πr.

Since C = 2πr, and since we already said the radius of the circle equals one radian, setting r = 1 means that C = 2π radians. In other words, there are 2π radians in the circumference of the circle, which is 360 degrees. So, 360 degrees = 2π radians. Knowing that, we can now convert between radians and degrees.

Since there are 2π radians in 360 degrees, we get: 2π rad = 360 deg. Diving both sides by 2π, rad = 360/2π = 180/π. And, because 360 deg = 2π rad: dividing both sides by 360, deg = 2π/360 = π/180. To summarize:
  • Given degrees, you get radians with rad = deg × 180/π.
  • Given radians, you get degrees = rad × π/180.
Radians for common degrees

You'll see charts that tell you 90 degrees equals π/2 radians, or 315 degrees equals 7π/4 radians. How did they get that?

Remember that C = 2π, meaning that there are 2π radians in 360 degrees. To find out how many radians are in, for example, 90 degrees, we just multiply 2π by the ratio 90/360, like this:

radians = 90/360 × 2π.

Reducing the fraction, we get ¼ × 2π, which simplifies to 2π/4, or π/2. So 90 degrees = π/2 radians.

Let's do a few more:

180 degrees = 180/360 × 2π = ½ × 2π = 2π/2 = π radians.
270 degrees = 270/360 × 2π = ¾ × 2π = 6π/4 = 3π/2 radians.
360 degrees = 360/360 × 2π = 2π radians.

Try the calculations for yourself! 45, 135, 225, and 315 degrees are all common angles.

Cheaper Gas Calculator (spreadsheet)

Is is worth driving to a more distant gas station to buy gas that's slightly cheaper? Here's a spreadsheet to find out. Just copy and paste into cell A1 in your favorite spreadsheet program, then enter your own data in cells A1 through A6. Your savings in cells A14 and A15 should be a positive number to make the trip worth while.

In cells A5 and A6 just put the one-way distance to the away station. The spreadsheet automatically calculates the round-trip cost.

Tested in Excel, OpenOffice Calc and Gnumeric.

15tank size (gallons)
29.55miles per gallon
3.619local price per gallon
3.599away price per gallon
2.00miles to local station (one way)
5.00miles to away station (one way)
  
=A1*A3cost to fill up at local station
=A1*A4cost to fill up at away station
=A8-A9savings from cheaper gas
  
=(A6-A5)*2extra round trip distance to away station
=A12/A2extra gallons to drive to cheaper station and back
=A13/A3extra travel cost to away station (based on local gas price)
  
=A10-A14savings (loss)
=A16/A1savings (loss) per gallon

Highlighting rows in Excel having specific text

We have an Excel spreadsheet and we'd like to somehow highlight cells containing specific text.  For example, we have the following data in cells A1 through A10 and we want to highlight any row containing the letter 'F', in this case cells A3, A5 and A8.

ABC
BCD
DEF
GHI
EFG
JKL
CDE
FGH
KLM
HIJ

First we need a formula that allows us to identify whether a cell contains the target text.  We can use the Search() function, like this: =SEARCH("F",A1).  Search returns the position of the first letter in the search string if it is found, otherwise it returns a #Value error.  This works well, but we are not interested in the position of the text, just whether or not it is found.  We also need to deal with the #Value error when the text is not found.  Here's how:

=NOT(ISERROR(SEARCH("F",A1)))

This formula will return TRUE if the text is found, or FALSE if it is not.  Now all we need is a way to highlight the cells containing the search text.  Enter conditional formatting!

Move the cursor to cell A1 and hit Formatting --> Contitional formatting in the menu.  Change the dropdown from "Cell value is" to "Formula is".  Then paste your formula in the formula box and set the background to red.  This will have the effect of setting the cell background to red if the cell contains the text we are looking for.  Hit OK to save.

Now hit the paintbrush button in the toolbar to copy this format to all the other cells.  You should see the cells with the letter F highlighted in red.

Detecting Dates in Microsoft Excel Spreadsheets

Microsoft Excel does not have an "isdate()" function, making it hard to determine whether a cell contains a date.

Excel VBA provides a function, but a formulaic solution also exists.  The trick is to realize that:
  1. Dates are stored as numbers in Excel, and
  2. You can use the Cell() function to return the format of a cell.
Armed with these two pieces of information, we can write a formula that returns TRUE (or one) if:
  1. The cell contains a number, and
  2. Is formatted as a date.
Here's how.

Use the IsNumber() function to determine whether or not the value contained in a cell is a number, like this: =ISNUMBER(A1).

Use the Cell() function to get the format of a cell, like this: =CELL("format",A1).

If Cell() returns a result in the range "D1" to "D5", the target cell -- in this case A1 --  is formatted as a date.  Therefore, we can write: =OR(CELL("format",A6)={"D1","D2","D3","D4","D5"}).

Or, more compactly: =LEFT(CELL("format",A6),1)="D".

Now we just need to combine the two formulas with an And(), like this:
=IF(AND(ISNUMBER(A5),LEFT(CELL("format",A5),1)="D"),1,0).

This formula returns one if the cell is a number and it is formatted as a date, otherwise it returns a zero.

Anonymous Inner (Lambda) Functions with PHP

<?php
$str = "hello world!";


/* 1. won't work with with anonymous inner (lambda) function before PHP 5.3
  $str = preg_replace_callback(
    '/world/',
    function( '$match', 'return "michael";' ),
    $str ); */
 

/* 2. use create_function instead */
$str = preg_replace_callback(
  '/world/',
  create_function( '$match', 'return "michael";' ),
  $str );
 

echo $str ; // output: hello michael!
?>