output formatting - How to build a function object like the built-ins such as Interpolation?

There are many built-in functions that return a function object, such as Interpolation[], BSplineFunction[] ,LinearSolveFunction[] and so on.

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Now given that I want to build a function called CAGDBSplineFunction[] like the built-in BSplineFunction[] with the help of Cox-DeBoor algorithm.

First trial, please see here

To achieve the dynamic effect and check the validity of the option like the built-in BSplineFunction

I refactored it as below:

CAGDBSplineFunction::invknots = 
  "Value of option SplineKnots \[Rule] `1` should be a non-decreasing \
   real sequence of length `2`, or a symbol Automatic.";

Options[CAGDBSplineFunction] = 
  {SplineDegree -> Automatic, SplineKnots -> Automatic};

CAGDBSplineFunction /: 
 MakeBoxes[CAGDBSplineFunction[pts_, opts : OptionsPattern[]], _] := 
 Module[{n, sk, sd, range},

  n = Length@pts - 1;
  sk = OptionValue[SplineKnots];
  sd = OptionValue[SplineDegree];
  (*check the validity of the option SplineKnots*)
  If[sk =!= Automatic,
   If[n + 1 + sd != Length[sk] - 1, 
    Message[CAGDBSplineFunction::invknots, sk, n + 2 + sd];
    Return[$Failed]];
    range = Through[{First, Last}@sk],
   range = {0, 1}

  ]
 ]
 InterpretationBox[
   RowBox[{"CAGDBSplineFunction", "[", "{", #1, ",", #2, "}", ",", 
   "\"<>\"", "]"}], CAGDBSplineFunction[pts, opts]] & @@ range
]

TEST

pts = {{0, 0}, {1, 1}, {2, -1}, {3, 0}, {4, -2}, {5, 1}};
f = CAGDBSplineFunction[pts]

However, it seems that the option value cannot be achieved in a MakeBoxes construct.

J.M. gives me the following suggestion:

Don't try to do both display and processing at once. Set a definition for evaluating CAGDBSplineFunction[], and then set a definition for displaying, via MakeBoxes

According to the J.M.'s hint, I add the defintion to CAGDBSplineFunction[]

CAGDBSplineFunction /: 
 MakeBoxes[CAGDBSplineFunction[pts_, opts : OptionsPattern[]], _] := 

  InterpretationBox[
    RowBox[{"CAGDBSplineFunction", "[", "{", #1, ",", #2, "}", ",", 
  "\"<>\"", "]"}], CAGDBSplineFunction[pts, opts]] & @@ 
    CAGDBSplineFunction[pts, opts]

CAGDBSplineFunction[pts_, opts : OptionsPattern[]] :=
 Module[
  {n, sk, sd, range},
  n = Length@pts - 1;
  sk = OptionValue[SplineKnots];

  sd = OptionValue[SplineDegree] /. Automatic -> 3;
  (*check the validity of the option SplineKnots*)
  If[sk =!= Automatic,
   If[n + 1 + sd != Length[sk] - 1, 
    Message[CAGDBSplineFunction::invknots, sk, n + 2 + sd];
    Return[$Failed]];
    range = Through[{First, Last}@sk],
   range = {0, 1}
  ]
 ]