In this Lab Assignment, which is one of the options for Lab 4, you will design and implement a simple graphical calculator. Like the previous labs, this lab assignment consists of two parts: in Part I, you will implement the parts of your program that have to do with expressions. In part II, you will implement the graphical part of your program.
To do the graphical part in a platform-independent, but still fairly simple way, we will use a web browser for the user interface. To prepare for this, and avoid last-minute problems, you might want to take a look at the introduction to Part II even before you start working on Part I.
Hints
Some assignments have hints. Often, these involve particular standard Haskell functions that you could use. Some of these functions are defined in modules that you have to import yourself explicitly. You can use the following resources to find more information about those functions:
Hoogle, the library function search engine.
Haskell Library Structure, many standard modules for you to browse.
A Graphical Calculator
The graphical calculator will be implemented as a web page that might look like this:
The page consists of a drawing area, and a text entry field below it. The user can type mathematical expressions in the text entry field, which, after pressing the Draw graph button (or maybe the Enter key on the keyboard), will be shown graphically in the drawing area.
Part I
In this part, you are going to design a datatype for modelling mathematical expressions containing a variable x. For example:- 3*x + 17.3
- sin x + cos x
- sin (2*x + 3.2) + 3*x + 5.7
- Numbers; these can be integers as well as floating point numbers
- Variables; there is only one variable, x
- Operators; for now it is enough with + and *.
- Functions; for now it is enough with sin and cos.
Put the answers for Part I in a module called Expr.hs
.
Assignments
Expr
that represents expressions of the above
kind.
You may represent integer numbers by floating point numbers; it is not necessary to have
two different constructor functions for this. For variables,
since the variable name is always x
,
the representation of variables does not need to include the variable name.
Your data type should be designed to make it easy to add more functions and more binary operators to the language.
that converts any expression to string. The strings that are produced should look something like the example expressions shown earlier. Use as little parentheses as possible, but consider that we want to be able to read an expression back without loss of information, as discussed in assignment D and E below.showExpr :: Expr -> String
It is not required to show floating point numbers that represent integer numbers without the decimal part. For example, you may choose to always show 2.0 as "2.0" and not as "2". (But you are allowed to do this.)
that, given an expression, and the value for the variable x, calculates the value of the expression.eval :: Expr -> Double -> Double
that, given a string, tries to interpret the string as an expression, and returns Just of that expression if it succeeds. Otherwise, Nothing will be returned. You are required to use an unmodified version of the module Parsing.hs to construct this function. See the lectures from Week 4 on parsing expressions.readExpr :: String -> Maybe Expr
The next assignment is about checking that your definition of
readExpr
matches
up with your definition of showExpr
.
One could define a property that simply
checks that, for any expression e1, if you show it, and then read it back in
again as an expression e2, then e1 and e2 should be the same.
However, this doesn't work if there is certain information loss in showing an expression. This happens, for example, if the expressions (1+2)+3 and 1+(2+3), which have different representations in your datatype (and are not equal), are both shown as "1+2+3".
A simple solutions to this problem is to include the parentheses when showing
1+(2+3) and only omit them when showing (1+2)+3. A nicer solution is
perhaps to use an equality test that takes the associativity of operators
into account (e.g. it considers 1+(2+3) and (1+2)+3 equal).
See the function assoc
in
exercise 3B from week 5 for an
example of how you could implement this.
that says that first showing and then reading an expression (using your functionsprop_ShowReadExpr :: Expr -> Bool
showExpr
and
readExpr
)
should produce "the same" result as the expression you started with.
Also define a generator for expressions:
Do not forget to take care of the size argument in the generator.arbExpr :: Int -> Gen Expr
Make Expr
an instance of the class
Arbitrary
and QuickCheck the result!
instance Arbitrary Expr where arbitrary = sized arbExpr
quickCheck
will call arbExpr
with sizes in the range [0..99]
, so make sure
arbExpr
produces reasonably sized expressions for testing
with this range of sizes.
which simplifies expressions so that subexpressions not involving variables are always simplified to their smallest representation, and that (sub)expressions representing x + 0 , 0 * x and 1 * x and similar terms are always simplified. Define (and run) QuickCheck properties that check that the simplifier is correct (e.g. 1+1 does not simplify to 3), and that it simplifies as much as possible (or more accurately: as much as you expect!).simplify :: Expr -> Expr
which differentiates the expression (with respect to x). You should use the simplify function to simplify the result.differentiate :: Expr -> Expr
Hints
To design the datatype, you might be inspired by the
Expr
datatype discussed in the lectures in Week 4. Read the slides and look at the example code! It is important that your datatype is simple and elegant; try not to use too many data constructors for example. If two different cases can be effectively expressed using one constructor, then do that.When designing your datatype
Expr
, think carefully about how you want to express the variable x. You may get inspired by theExpr
datatype from the lectures, but remember that there is a difference; in your datatype you only have to represent one variable, called x, whereas in the lecture we allowed for several different variables.When showing and reading expressions, we have to decide where we allow and require parentheses. Parentheses are required only in the following cases:
- When the arguments of a *-expression use +. For example: (3.1+4.2)*7
- When the argument of sin or cos uses * or +. For example: sin (3.2*x)
In all other cases, you should not require parentheses. For example:
- Allow 2*3+4*5 instead of 2*3+(4*5)
- Allow sin x instead of sin(x)
- Allow sin cos x instead of sin (cos x)
- Allow sin x + cos x instead of (sin x) + (cos x)
Make sure that the above expressions are all parsed correctly by your program!
For the functions
eval
,showExpr
,readExpr
,arbExpr
, andprop_ShowReadExpr
also consult the lectures from Week 4, and look at the example code that is provided.For the function
readExpr
, to be able to parse floating point numbers (Doubles) and sin and cos, you only have to change the parser for factors.To parse floating point numbers, make use of the standard Haskell functions
reads
exported asreadsP
in the Parsing module. Use it on the right type (Double
), and see what happens!read "17.34" :: Double
...read "17.34cykel" :: Double
...reads "17.34cykel" :: [(Double,String)]
...-
If you have a hard time understanding the generated counter examples for your
property, it is probably a good idea to let Haskell derive the show function
for your
Expr
datatype, instead of making your own instance of Show. So, usederiving Show
on your expression datatype while testing!
Part II
Creating graphical user interfaces with Haskell
In this part, you are going to implement the graphical part of the calculator. To do this in a platform-independent way, we will use a web browser for the user interface.threepenny-gui is a Haskell library for creating web pages with interactive parts (buttons, sliders, text entry boxes, etc). To install it, you can use the cabal tool (which you already have, if you have installed the Haskell Platform) in a terminal window:
cabal install threepenny-gui
To make things easier, you can use the following two modules:
- ThreepennyPages.hs provides some helper functions for creating user interface elements.
- Calculator.hs provides a starting point for implementing the graphical part of the calculator.
Structure of the User Interface
The graphical interface consists of at least two parts: (1) the drawing area, where the function is going to be drawn, and (2) the text entry field.
The drawing area is a "canvas" element of a certain size (you decide yourself, but let us suppose it has width and height of 300). A canvas has a coordinate system that works in pixels. Here is how it works:
(0,0) | (300,0) |
(0,300) | (300,300) |
Perhaps surprising is that y-coordinates are upside down; they are 0 at the top, and 300 at the bottom.
The tricky thing using this canvas to draw our functions is that the coordinate system we are used to in mathematics has (0,0) in the middle, and the y-coordinates are not upside down. For example, the coordinate system for our functions might work like this:
(-6.0,6.0) | (6.0,6.0) | |
(0,0) | ||
(-6.0,-6.0) | (6.0,-6.0) |
So, some conversion is needed between pixels and mathematical coordinates. Note, though, that both coordinate systems are represented using the type
Double
.
Assignments
The answers for Part II should be put in a module calledCalculator.hs
.
It should of course import the module Expr
that you created in Part I.
In this assignment you are free to be creative when designing the calculator. But for those who just want to focus on the important stuff, we have prepared a stub program that constructs the web page for you: Calculator.hs, which uses ThreepennyPages.hs mentioned above. These files serve as examples of how to use threepenny-gui, so it might be useful to look at them even if you decide not to use them directly.
The function gets three arguments;points :: Expr -> Double -> (Int,Int) -> [Point]
points exp scale (width,height)
:
- An expression
exp
, - A scaling value
scale
, - The
width
andheight
of the drawing area
Point
is defined in threepenny-gui
and is just a pair of numbers:
The idea is thattype Point = (Double, Double)
points
will calculate all the points of the graph in terms
of pixels. The scaling value tells you the ratio between pixels and floating
point numbers. The arguments width
and height
tell you how big the drawing area
is. We assume that the origin (0,0) point is in the middle of our drawing area.
For the canvas and function coordinate systems above, we would use 0.04 for the scale (since 1 pixel corresponds to 0.04 in the floating point world, this is (6.0 + 6.0) / 300), and (300,300) for the width and height.
If you are basing your code on the provided module Calculator.hs, this part only involves completing the function
which reads the expression from the given input element and draws the graph on the given canvas.readAndDraw :: Element -> Canvas -> UI ()
When the user types in something that is not an expression, you may decide yourself what to do. The easiest thing is to draw nothing.
Make sure there is a way to zoom out after you have zoomed in.
The recommended solution is to let the differentiated expression
replace the expression in the text entry field.
This allows you to differentiate
many times, and provides a nice way to test that showExpr
and simplify
work as expected.)
Hints
To convert back and forth between Ints and Doubles, the following function might come in handy:
These functions have more general types than the ones given above. For other conversion functions, use Hoogle!fromIntegral :: Int -> Double round :: Double -> Int
To implement the function points
,
it is probably a good idea to define
the following two local helper functions:
where – converts a pixel x-coordinate to a real x-coordinatepixToReal :: Double -> Double
pixToReal x = ... – converts a real y-coordinate to a pixel y-coordinaterealToPix :: Double -> Double
realToPix y = ...
The easiest way to draw the graph on the canvas is to use the function
path
defined in ThreepennyPages.hs
which connects a sequence of points with lines. The first argument is a web colour name (e.g.path :: String -> [Point] -> Canvas -> UI ()
"black"
,
"#000"
,
"blue"
,
"#00f"
).
Extra Assignments
You may freely extend your solution to include other features, but if possible do so in a way that does not make it more difficult to understand the code for the parts of the lab described above.One idea:
- Add a way for the user to enter multiple functions and drawn them in different colours.
Submission
Submit your solutions using the Fire system.Your submission should consist of the following files:
Expr.hs
Calculator.hs
Before you submit your code, spend some time on cleaning up your code; make it simpler, remove unneccessary things, etc. We will reject your solution if it is not clean. Clean code:
- Does not have long lines (< 78 characters)
- Has a consistent layout
- Has type signatures for all top-level functions
- Has good comments
- Has no junk (junk is unused code, commented code, unneccessary comments)
- Has no overly complicated function definitions
- Does not contain any repetitive code (copy-and-paste programming)
This year we have added Automatic checks in Fire that will run when you submit your answers. The tests include runing hlint and perhaps testing some of your functions with QuickCheck. The purpose of this is to give you some quick feedback and to help speed up the grading process. If the feedback you get is not helpful, you can safely ignore it.
Good Luck!