A.3.3.CC:4.3 - Choose a representation that supports the operation
Compare the following forms using the operation the reader actually needs.
| Form | How to construct and use it | When it becomes inconvenient |
|---|---|---|
| Variables with implicit constraints | Keep the participant values and equations or predicates between them. Test a proposed tuple by substitution. | Finding a satisfying tuple can require solving coupled conditions. |
| Parametrization | Choose parameters u and a reconstruction q = f(u) that satisfies the constraints. Perform the calculation in u and reconstruct the required participant values. | The chosen parameters may cover only part of the admitted set, repeat a configuration or become singular. |
| Explicit finite set | Generate candidate tuples, retain those satisfying the conditions and inspect or compare the resulting set. | The product of value sets can become too large to enumerate. |
For a parametrization, establish the coverage its use needs. If the task claims to represent every admitted configuration, explain how each can be obtained, possibly using several charts or cases. State when two parameter values represent the same configuration. A full turn of an angle repeats an orientation; treating its endpoints as unrelated can break a continuity calculation.
Keep redundant coordinates when they preserve useful structure or avoid a difficult elimination. If the task instead uses a local count of freely variable coordinates, establish independence and regularity of the active equality constraints before subtracting their number. A count alone supplies neither a parametrization nor its global coverage.
When replacing one representation with another, reconstruct the participant values and compare the receiving result. C.29.1 supplies a fuller transfer comparison when the replacement can change an operation or lose information. Group configurations as equivalent only when every member gives the same required answer and the required operations preserve that grouping. This mathematical construction is called a quotient. For an operation on the groups, check that choosing another member of the same group gives an equivalent result. Keep differently labeled endpoints distinct when exchanging them changes the answer.