The circle–ellipse problem in software development (sometimes called the square–rectangle problem) illustrates several pitfalls which can arise when using subtype polymorphism in object modelling. The issues are most commonly encountered when using object-oriented programming (OOP). By definition, this problem is a violation of the Liskov substitution principle, one of the SOLID principles.
The problem concerns which subtyping or inheritance relationship should exist between classes which represent circles and ellipses (or, similarly, squares and rectangles). More generally, the problem illustrates the difficulties which can occur when a base class contains methods which mutate an object in a manner which may invalidate a (stronger) invariant found in a derived class, causing the Liskov substitution principle to be violated.
The existence of the circle–ellipse problem is sometimes used to criticize object-oriented programming. It may also imply that hierarchical taxonomies are difficult to make universal, implying that situational classification systems may be more practical.
Description It is a central tenet of object-oriented analysis and design that subtype polymorphism, which is implemented in most object-oriented languages via inheritance, should be used to model object types that are subsets of each other; this is commonly referred to as the is-a relationship. In the present example, the set of circles is a subset of the set of ellipses; circles can be defined as ellipses whose major and minor axes are the same length. Thus, code written in an object-oriented language that models shapes will frequently choose to make a subclass of , i.e. inheriting from it.
A subclass must provide support for all behaviour supported by the super-class; subclasses must implement any mutator methods defined in a base class. In the present case, the method alters the length of one of its axes in place. If inherits from , it must also have a method , but the result of this method would be to change a circle into something that is no longer a circle. The class cannot simultaneously satisfy its own invariant and the behavioural requirements of the method.
A related problem with this inheritance arises when considering the implementation. An ellipse requires more states to be described than a circle, because the former needs attributes to specify the length and rotation of the major and minor axes whereas a circle needs only a radius. It may be possible to avoid this if the language (such as Eiffel) makes constant values of a class, functions without arguments, and data members interchangeable.
Some authors have suggested reversing the relationship between circle and ellipse, on the grounds that an ellipse is a circle with more abilities. Unfortunately, ellipses fail to satisfy many of the invariants of circles; if has a method , must now provide it, too.
Possible solutions One may solve the problem by: changing the model using a different language (or an existing or custom-written extension of some existing language) using a different paradigm Exactly which option is appropriate will depend on who wrote and who wrote . If the same author is designing them both from scratch, then the author will be able to define the interface to handle this situation. If the object was already written, and cannot be changed, then the options are more limited.
Change the model
Return success or failure value Allow the objects to return a "success" or "failure" value for each modifier or raise an exception on failure. This is usually done in the case of file I/O, but can also be helpful here. Now, works, and returns "true", while simply returns "false". This is in general good practice, but may require that the original author of anticipated such a problem, and defined the mutators as returning a value. Also, it requires the client code to test the return value for support of the stretch function, which in effect is like testing if the referenced object is either a circle or an ellipse. Another way to look at this is that it is like putting in the contract that the contract may or may not be fulfilled depending on the object implementing the interface. Eventually, it is only a clever way to bypass the Liskov constraint by stating up-front that the post condition may or may not be valid.
Alternately, could throw an exception (but depending on the language, this may also require that the original author of declare that it may throw an exception).
Return the new value of X This is a similar solution to the above, but is slightly more powerful. now returns the new value of its X dimension. Now, can simply return its current radius. All modifications must be done through , which preserves the circle invariant.
Allow for a weaker contract on Ellipse If the interface contract for states only that "stretchX modifies the X axis", and does not state "and nothing else will change", then could simply force the X and Y dimensions to be the same. and both modify both the X and Y size.
Convert the Circle into an Ellipse If is called, then changes itself into an . For example, in Common Lisp, this can be done via the method. This may be dangerous, however, if some other function is expecting it to be a . Some languages preclude this type of change, and others impose restrictions on the class to be an acceptable replacement for . For languages that allow implicit conversion like C++, this may only be a partial solution solving the problem on call-by-copy, but not on call-by-reference.
Make all instances constant One can change the model so that instances of the classes represent constant values (i.e., they are immutable). This is the implementation that is used in purely functional programming.
In this case, methods such as must be changed to yield a new instance, rather than modifying the instance they act on. This means that it is no longer a problem to define , and the inheritance reflects the mathematical relationship between circles and ellipses.
A disadvantage is that changing the value of an instance then requires an assignment, which is inconvenient and prone to programming errors, e.g.,
A second disadvantage is that such an assignment conceptually involves a temporary value, which could reduce performance and be difficult to optimise.
Factor out modifiers One can define a new class , and put the modifiers from in it. The only inherits queries from .
This has a disadvantage of introducing an extra class where all that is desired is specify that does not inherit modifiers from .
Impose preconditions on modifiers One can specify that is only allowed on instances satisfying , and will otherwise throw an exception. This requires anticipation of the problem when Ellipse is defined.
Factor out common functionality into an abstract base class Create an abstract base class called and put methods that work with both s and s in this class. Functions that can deal with either type of object will expect an , and functions that use - or -specific requirements will use the descendant classes. However, is then no longer an subclass, leading to the "a is not a sort of " situation described above.
Drop all inheritance relationships This solves the problem at a stroke. Any common operations desired for both a Circle and Ellipse can be abstracted out to a common interface that each class implements, or into mixins.
Also, one may provide conversion methods like , which returns a mutable Ellipse object initialized using the circle's radius. From that point on, it is a separate object and can be mutated separately from the original circle without issue. Methods converting the other way need not commit to one strategy. For instance, there can be both and , and any other strategy desired.
Combine class Circle into class Ellipse Then, wherever a circle was used before, use an ellipse.
A circle can already be represented by an ellipse. There is no reason to have class unless it needs some circle-specific methods that can't be applied to an ellipse, or unless the programmer wishes to benefit from conceptual and/or performance advantages of the circle's simpler model.
Inverse inheritance Majorinc proposed a model that divides methods on modifiers, selectors and general methods. Only selectors can be automatically inherited from superclass, while modifiers should be inherited from subclass to superclass. In general case, the methods must be explicitly inherited. The model can be emulated in languages with multiple inheritance, using abstract classes.Kazimir Majorinc, Ellipse-Circle Dilemma and Inverse Inheritance, ITI 98, Proceedings of the 20th International Conference of Information Technology Interfaces, Pula, 1998
Change the programming language This problem has straightforward solutions in a sufficiently powerful OO programming system. Essentially, the circle–ellipse problem is one of synchronizing two representations of type: the de facto type based on the properties of the object, and the formal type associated with the object by the object system. If these two pieces of information, which are ultimately only bits in the machine, are kept synchronized so that they say the same thing, everything is fine. It is clear that a circle cannot satisfy the invariants required of it while its base ellipse methods allow mutation of parameters. However, the possibility exists that when a circle cannot meet the circle invariants, its type can be updated so that it becomes an ellipse. If a circle which has become a de facto ellipse doesn't change type, then its type is a piece of information which is now out of date, reflecting the history of the object (how it was once constructed) and not its present reality (what it has since mutated into).
Many object systems in popular use are based on a design which takes it for granted that an object carries the same type over its entire lifetime, from construction to finalization. This is not a limitation of OOP, but rather of particular implementations only.
The following example uses the Common Lisp Object System (CLOS) in which objects can change class without losing their identity. All variables or other storage locations which hold a reference to an object continue to hold a reference to that same object after it changes class.
The circle and ellipse models are deliberately simplified to avoid distracting details which are not relevant to the circle–ellipse problem. An ellipse has two semi-axes called and in the code. Being an ellipse, a circle inherits these, and also has a property, which value is equal to that of the axes (which must, of course, be equal to each other).
(defgeneric check-constraints (shape))
;; The accessors on shape objects. Constraints on objects ;; need to be checked after either axis value is set. (defgeneric h-axis (shape)) (defgeneric (setf h-axis) (new-value shape) (:method :after (new-value shape) (check-constraints shape))) (defgeneric v-axis (shape)) (defgeneric (setf v-axis) (new-value shape) (:method :after (new-value shape) (check-constraints shape)))
(defclass ellipse () ((h-axis :type real :accessor h-axis :initarg :h-axis) (v-axis :type real :accessor v-axis :initarg :v-axis)))
(defclass circle (ellipse) ((radius :type real :accessor radius :initarg :radius)))
;;; A circle has a radius, but also a h-axis and v-axis that ;;; it inherits from an ellipse. These must be kept in sync ;;; with the radius when the object is initialized and ;;; when those values change.
(defmethod initialize-instance :after ((c circle) &key radius) (setf (radius c) radius)) ;; via the setf method below
(defmethod (setf radius) :after ((new-value real) (c circle)) ;; We use SLOT-VALUE, rather than the accessors, to avoid changing ;; class unnecessarily between the two assignments; as the circle ;; will have different h-axis and v-axis values between the ;; assignments, and then the same values after assignments.