The widely used B-spline method is limited by its incapability to realize high-order continuity and high local adjustment simultaneously. The shape of the B-spline curve and surface is uniquely determined by the control points and knot vector. These factors influence the design effects and convenience of the B-spline method. This study aims to overcome these limitations. We construct a new curve and surface that possess the high order continuity of the high-order B-spline method

the high local adjustment of the low-order B-spline method

and the shape adjustment property of the rational B-spline method as determined by the weight factor. We first construct a set of blending functions with a parameter on the trigonometric function space. We then define the new curve and tensor product surface

which have the same structure as those of the cubic B-spline curve and surface

respectively. The new curve and surface inherit many of the B-spline method properties

such as convex hull

symmetry

and geometric invariability. The difference is that although both the new and cubic B-spline curves are based on a four-point segment

the latter is C continuous

while for equidistant knots

the new curve is generally C continuous and can reach C continuity when taking a special parameter. A new curve that is C continuous has one shape parameter

which can be used to adjust the shape of the curve without changing the control points. In addition

taking the special parameter

the new curve can automatically interpolate the given set of points. The new surface has properties corresponding to the new curve. This study presents a kind of shape-adjustable piecewise combination curve and surface

which can achieve high-order continuity under strong local control capability. These features can be used for the possible design of a high-order smooth curve and surface. The new curve realizes the uniform representation of approximation and interpolation. The construction method of the blending function is general and can be used to construct other functions with similar properties.